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Polytope of Type {4,4,4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,4,4}*512
Also Known As : {{4,4|2},{4,4|2},{4,4|2}}. if this polytope has another name.
Group : SmallGroup(512,554852)
Rank : 5
Schlafli Type : {4,4,4,4}
Number of vertices, edges, etc : 4, 8, 8, 8, 4
Order of s0s1s2s3s4 : 4
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,4,4,4}*256, {4,4,4,2}*256, {4,2,4,4}*256, {4,4,2,4}*256
   4-fold quotients : {2,2,4,4}*128, {2,4,4,2}*128, {4,4,2,2}*128, {2,4,2,4}*128, {4,2,2,4}*128, {4,2,4,2}*128
   8-fold quotients : {2,2,2,4}*64, {2,2,4,2}*64, {2,4,2,2}*64, {4,2,2,2}*64
   16-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1, 33)(  2, 34)(  3, 35)(  4, 36)(  5, 37)(  6, 38)(  7, 39)(  8, 40)
(  9, 41)( 10, 42)( 11, 43)( 12, 44)( 13, 45)( 14, 46)( 15, 47)( 16, 48)
( 17, 49)( 18, 50)( 19, 51)( 20, 52)( 21, 53)( 22, 54)( 23, 55)( 24, 56)
( 25, 57)( 26, 58)( 27, 59)( 28, 60)( 29, 61)( 30, 62)( 31, 63)( 32, 64)
( 65, 97)( 66, 98)( 67, 99)( 68,100)( 69,101)( 70,102)( 71,103)( 72,104)
( 73,105)( 74,106)( 75,107)( 76,108)( 77,109)( 78,110)( 79,111)( 80,112)
( 81,113)( 82,114)( 83,115)( 84,116)( 85,117)( 86,118)( 87,119)( 88,120)
( 89,121)( 90,122)( 91,123)( 92,124)( 93,125)( 94,126)( 95,127)( 96,128)
(129,161)(130,162)(131,163)(132,164)(133,165)(134,166)(135,167)(136,168)
(137,169)(138,170)(139,171)(140,172)(141,173)(142,174)(143,175)(144,176)
(145,177)(146,178)(147,179)(148,180)(149,181)(150,182)(151,183)(152,184)
(153,185)(154,186)(155,187)(156,188)(157,189)(158,190)(159,191)(160,192)
(193,225)(194,226)(195,227)(196,228)(197,229)(198,230)(199,231)(200,232)
(201,233)(202,234)(203,235)(204,236)(205,237)(206,238)(207,239)(208,240)
(209,241)(210,242)(211,243)(212,244)(213,245)(214,246)(215,247)(216,248)
(217,249)(218,250)(219,251)(220,252)(221,253)(222,254)(223,255)(224,256)
(257,289)(258,290)(259,291)(260,292)(261,293)(262,294)(263,295)(264,296)
(265,297)(266,298)(267,299)(268,300)(269,301)(270,302)(271,303)(272,304)
(273,305)(274,306)(275,307)(276,308)(277,309)(278,310)(279,311)(280,312)
(281,313)(282,314)(283,315)(284,316)(285,317)(286,318)(287,319)(288,320)
(321,353)(322,354)(323,355)(324,356)(325,357)(326,358)(327,359)(328,360)
(329,361)(330,362)(331,363)(332,364)(333,365)(334,366)(335,367)(336,368)
(337,369)(338,370)(339,371)(340,372)(341,373)(342,374)(343,375)(344,376)
(345,377)(346,378)(347,379)(348,380)(349,381)(350,382)(351,383)(352,384)
(385,417)(386,418)(387,419)(388,420)(389,421)(390,422)(391,423)(392,424)
(393,425)(394,426)(395,427)(396,428)(397,429)(398,430)(399,431)(400,432)
(401,433)(402,434)(403,435)(404,436)(405,437)(406,438)(407,439)(408,440)
(409,441)(410,442)(411,443)(412,444)(413,445)(414,446)(415,447)(416,448)
(449,481)(450,482)(451,483)(452,484)(453,485)(454,486)(455,487)(456,488)
(457,489)(458,490)(459,491)(460,492)(461,493)(462,494)(463,495)(464,496)
(465,497)(466,498)(467,499)(468,500)(469,501)(470,502)(471,503)(472,504)
(473,505)(474,506)(475,507)(476,508)(477,509)(478,510)(479,511)(480,512);;
s1 := (  1,273)(  2,274)(  3,275)(  4,276)(  5,277)(  6,278)(  7,279)(  8,280)
(  9,281)( 10,282)( 11,283)( 12,284)( 13,285)( 14,286)( 15,287)( 16,288)
( 17,257)( 18,258)( 19,259)( 20,260)( 21,261)( 22,262)( 23,263)( 24,264)
( 25,265)( 26,266)( 27,267)( 28,268)( 29,269)( 30,270)( 31,271)( 32,272)
( 33,306)( 34,305)( 35,308)( 36,307)( 37,310)( 38,309)( 39,312)( 40,311)
( 41,314)( 42,313)( 43,316)( 44,315)( 45,318)( 46,317)( 47,320)( 48,319)
( 49,290)( 50,289)( 51,292)( 52,291)( 53,294)( 54,293)( 55,296)( 56,295)
( 57,298)( 58,297)( 59,300)( 60,299)( 61,302)( 62,301)( 63,304)( 64,303)
( 65,341)( 66,342)( 67,343)( 68,344)( 69,337)( 70,338)( 71,339)( 72,340)
( 73,349)( 74,350)( 75,351)( 76,352)( 77,345)( 78,346)( 79,347)( 80,348)
( 81,325)( 82,326)( 83,327)( 84,328)( 85,321)( 86,322)( 87,323)( 88,324)
( 89,333)( 90,334)( 91,335)( 92,336)( 93,329)( 94,330)( 95,331)( 96,332)
( 97,374)( 98,373)( 99,376)(100,375)(101,370)(102,369)(103,372)(104,371)
(105,382)(106,381)(107,384)(108,383)(109,378)(110,377)(111,380)(112,379)
(113,358)(114,357)(115,360)(116,359)(117,354)(118,353)(119,356)(120,355)
(121,366)(122,365)(123,368)(124,367)(125,362)(126,361)(127,364)(128,363)
(129,409)(130,410)(131,411)(132,412)(133,413)(134,414)(135,415)(136,416)
(137,401)(138,402)(139,403)(140,404)(141,405)(142,406)(143,407)(144,408)
(145,393)(146,394)(147,395)(148,396)(149,397)(150,398)(151,399)(152,400)
(153,385)(154,386)(155,387)(156,388)(157,389)(158,390)(159,391)(160,392)
(161,442)(162,441)(163,444)(164,443)(165,446)(166,445)(167,448)(168,447)
(169,434)(170,433)(171,436)(172,435)(173,438)(174,437)(175,440)(176,439)
(177,426)(178,425)(179,428)(180,427)(181,430)(182,429)(183,432)(184,431)
(185,418)(186,417)(187,420)(188,419)(189,422)(190,421)(191,424)(192,423)
(193,477)(194,478)(195,479)(196,480)(197,473)(198,474)(199,475)(200,476)
(201,469)(202,470)(203,471)(204,472)(205,465)(206,466)(207,467)(208,468)
(209,461)(210,462)(211,463)(212,464)(213,457)(214,458)(215,459)(216,460)
(217,453)(218,454)(219,455)(220,456)(221,449)(222,450)(223,451)(224,452)
(225,510)(226,509)(227,512)(228,511)(229,506)(230,505)(231,508)(232,507)
(233,502)(234,501)(235,504)(236,503)(237,498)(238,497)(239,500)(240,499)
(241,494)(242,493)(243,496)(244,495)(245,490)(246,489)(247,492)(248,491)
(249,486)(250,485)(251,488)(252,487)(253,482)(254,481)(255,484)(256,483);;
s2 := (  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 69)(  6, 70)(  7, 71)(  8, 72)
(  9, 73)( 10, 74)( 11, 75)( 12, 76)( 13, 77)( 14, 78)( 15, 79)( 16, 80)
( 17, 81)( 18, 82)( 19, 83)( 20, 84)( 21, 85)( 22, 86)( 23, 87)( 24, 88)
( 25, 89)( 26, 90)( 27, 91)( 28, 92)( 29, 93)( 30, 94)( 31, 95)( 32, 96)
( 33, 97)( 34, 98)( 35, 99)( 36,100)( 37,101)( 38,102)( 39,103)( 40,104)
( 41,105)( 42,106)( 43,107)( 44,108)( 45,109)( 46,110)( 47,111)( 48,112)
( 49,113)( 50,114)( 51,115)( 52,116)( 53,117)( 54,118)( 55,119)( 56,120)
( 57,121)( 58,122)( 59,123)( 60,124)( 61,125)( 62,126)( 63,127)( 64,128)
(129,193)(130,194)(131,195)(132,196)(133,197)(134,198)(135,199)(136,200)
(137,201)(138,202)(139,203)(140,204)(141,205)(142,206)(143,207)(144,208)
(145,209)(146,210)(147,211)(148,212)(149,213)(150,214)(151,215)(152,216)
(153,217)(154,218)(155,219)(156,220)(157,221)(158,222)(159,223)(160,224)
(161,225)(162,226)(163,227)(164,228)(165,229)(166,230)(167,231)(168,232)
(169,233)(170,234)(171,235)(172,236)(173,237)(174,238)(175,239)(176,240)
(177,241)(178,242)(179,243)(180,244)(181,245)(182,246)(183,247)(184,248)
(185,249)(186,250)(187,251)(188,252)(189,253)(190,254)(191,255)(192,256)
(257,321)(258,322)(259,323)(260,324)(261,325)(262,326)(263,327)(264,328)
(265,329)(266,330)(267,331)(268,332)(269,333)(270,334)(271,335)(272,336)
(273,337)(274,338)(275,339)(276,340)(277,341)(278,342)(279,343)(280,344)
(281,345)(282,346)(283,347)(284,348)(285,349)(286,350)(287,351)(288,352)
(289,353)(290,354)(291,355)(292,356)(293,357)(294,358)(295,359)(296,360)
(297,361)(298,362)(299,363)(300,364)(301,365)(302,366)(303,367)(304,368)
(305,369)(306,370)(307,371)(308,372)(309,373)(310,374)(311,375)(312,376)
(313,377)(314,378)(315,379)(316,380)(317,381)(318,382)(319,383)(320,384)
(385,449)(386,450)(387,451)(388,452)(389,453)(390,454)(391,455)(392,456)
(393,457)(394,458)(395,459)(396,460)(397,461)(398,462)(399,463)(400,464)
(401,465)(402,466)(403,467)(404,468)(405,469)(406,470)(407,471)(408,472)
(409,473)(410,474)(411,475)(412,476)(413,477)(414,478)(415,479)(416,480)
(417,481)(418,482)(419,483)(420,484)(421,485)(422,486)(423,487)(424,488)
(425,489)(426,490)(427,491)(428,492)(429,493)(430,494)(431,495)(432,496)
(433,497)(434,498)(435,499)(436,500)(437,501)(438,502)(439,503)(440,504)
(441,505)(442,506)(443,507)(444,508)(445,509)(446,510)(447,511)(448,512);;
s3 := (  1,129)(  2,130)(  3,131)(  4,132)(  5,133)(  6,134)(  7,135)(  8,136)
(  9,137)( 10,138)( 11,139)( 12,140)( 13,141)( 14,142)( 15,143)( 16,144)
( 17,153)( 18,154)( 19,155)( 20,156)( 21,157)( 22,158)( 23,159)( 24,160)
( 25,145)( 26,146)( 27,147)( 28,148)( 29,149)( 30,150)( 31,151)( 32,152)
( 33,161)( 34,162)( 35,163)( 36,164)( 37,165)( 38,166)( 39,167)( 40,168)
( 41,169)( 42,170)( 43,171)( 44,172)( 45,173)( 46,174)( 47,175)( 48,176)
( 49,185)( 50,186)( 51,187)( 52,188)( 53,189)( 54,190)( 55,191)( 56,192)
( 57,177)( 58,178)( 59,179)( 60,180)( 61,181)( 62,182)( 63,183)( 64,184)
( 65,195)( 66,196)( 67,193)( 68,194)( 69,199)( 70,200)( 71,197)( 72,198)
( 73,203)( 74,204)( 75,201)( 76,202)( 77,207)( 78,208)( 79,205)( 80,206)
( 81,219)( 82,220)( 83,217)( 84,218)( 85,223)( 86,224)( 87,221)( 88,222)
( 89,211)( 90,212)( 91,209)( 92,210)( 93,215)( 94,216)( 95,213)( 96,214)
( 97,227)( 98,228)( 99,225)(100,226)(101,231)(102,232)(103,229)(104,230)
(105,235)(106,236)(107,233)(108,234)(109,239)(110,240)(111,237)(112,238)
(113,251)(114,252)(115,249)(116,250)(117,255)(118,256)(119,253)(120,254)
(121,243)(122,244)(123,241)(124,242)(125,247)(126,248)(127,245)(128,246)
(257,385)(258,386)(259,387)(260,388)(261,389)(262,390)(263,391)(264,392)
(265,393)(266,394)(267,395)(268,396)(269,397)(270,398)(271,399)(272,400)
(273,409)(274,410)(275,411)(276,412)(277,413)(278,414)(279,415)(280,416)
(281,401)(282,402)(283,403)(284,404)(285,405)(286,406)(287,407)(288,408)
(289,417)(290,418)(291,419)(292,420)(293,421)(294,422)(295,423)(296,424)
(297,425)(298,426)(299,427)(300,428)(301,429)(302,430)(303,431)(304,432)
(305,441)(306,442)(307,443)(308,444)(309,445)(310,446)(311,447)(312,448)
(313,433)(314,434)(315,435)(316,436)(317,437)(318,438)(319,439)(320,440)
(321,451)(322,452)(323,449)(324,450)(325,455)(326,456)(327,453)(328,454)
(329,459)(330,460)(331,457)(332,458)(333,463)(334,464)(335,461)(336,462)
(337,475)(338,476)(339,473)(340,474)(341,479)(342,480)(343,477)(344,478)
(345,467)(346,468)(347,465)(348,466)(349,471)(350,472)(351,469)(352,470)
(353,483)(354,484)(355,481)(356,482)(357,487)(358,488)(359,485)(360,486)
(361,491)(362,492)(363,489)(364,490)(365,495)(366,496)(367,493)(368,494)
(369,507)(370,508)(371,505)(372,506)(373,511)(374,512)(375,509)(376,510)
(377,499)(378,500)(379,497)(380,498)(381,503)(382,504)(383,501)(384,502);;
s4 := (  1, 17)(  2, 18)(  3, 19)(  4, 20)(  5, 21)(  6, 22)(  7, 23)(  8, 24)
(  9, 25)( 10, 26)( 11, 27)( 12, 28)( 13, 29)( 14, 30)( 15, 31)( 16, 32)
( 33, 49)( 34, 50)( 35, 51)( 36, 52)( 37, 53)( 38, 54)( 39, 55)( 40, 56)
( 41, 57)( 42, 58)( 43, 59)( 44, 60)( 45, 61)( 46, 62)( 47, 63)( 48, 64)
( 65, 81)( 66, 82)( 67, 83)( 68, 84)( 69, 85)( 70, 86)( 71, 87)( 72, 88)
( 73, 89)( 74, 90)( 75, 91)( 76, 92)( 77, 93)( 78, 94)( 79, 95)( 80, 96)
( 97,113)( 98,114)( 99,115)(100,116)(101,117)(102,118)(103,119)(104,120)
(105,121)(106,122)(107,123)(108,124)(109,125)(110,126)(111,127)(112,128)
(129,145)(130,146)(131,147)(132,148)(133,149)(134,150)(135,151)(136,152)
(137,153)(138,154)(139,155)(140,156)(141,157)(142,158)(143,159)(144,160)
(161,177)(162,178)(163,179)(164,180)(165,181)(166,182)(167,183)(168,184)
(169,185)(170,186)(171,187)(172,188)(173,189)(174,190)(175,191)(176,192)
(193,209)(194,210)(195,211)(196,212)(197,213)(198,214)(199,215)(200,216)
(201,217)(202,218)(203,219)(204,220)(205,221)(206,222)(207,223)(208,224)
(225,241)(226,242)(227,243)(228,244)(229,245)(230,246)(231,247)(232,248)
(233,249)(234,250)(235,251)(236,252)(237,253)(238,254)(239,255)(240,256)
(257,273)(258,274)(259,275)(260,276)(261,277)(262,278)(263,279)(264,280)
(265,281)(266,282)(267,283)(268,284)(269,285)(270,286)(271,287)(272,288)
(289,305)(290,306)(291,307)(292,308)(293,309)(294,310)(295,311)(296,312)
(297,313)(298,314)(299,315)(300,316)(301,317)(302,318)(303,319)(304,320)
(321,337)(322,338)(323,339)(324,340)(325,341)(326,342)(327,343)(328,344)
(329,345)(330,346)(331,347)(332,348)(333,349)(334,350)(335,351)(336,352)
(353,369)(354,370)(355,371)(356,372)(357,373)(358,374)(359,375)(360,376)
(361,377)(362,378)(363,379)(364,380)(365,381)(366,382)(367,383)(368,384)
(385,401)(386,402)(387,403)(388,404)(389,405)(390,406)(391,407)(392,408)
(393,409)(394,410)(395,411)(396,412)(397,413)(398,414)(399,415)(400,416)
(417,433)(418,434)(419,435)(420,436)(421,437)(422,438)(423,439)(424,440)
(425,441)(426,442)(427,443)(428,444)(429,445)(430,446)(431,447)(432,448)
(449,465)(450,466)(451,467)(452,468)(453,469)(454,470)(455,471)(456,472)
(457,473)(458,474)(459,475)(460,476)(461,477)(462,478)(463,479)(464,480)
(481,497)(482,498)(483,499)(484,500)(485,501)(486,502)(487,503)(488,504)
(489,505)(490,506)(491,507)(492,508)(493,509)(494,510)(495,511)(496,512);;
poly := Group([s0,s1,s2,s3,s4]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(512)!(  1, 33)(  2, 34)(  3, 35)(  4, 36)(  5, 37)(  6, 38)(  7, 39)
(  8, 40)(  9, 41)( 10, 42)( 11, 43)( 12, 44)( 13, 45)( 14, 46)( 15, 47)
( 16, 48)( 17, 49)( 18, 50)( 19, 51)( 20, 52)( 21, 53)( 22, 54)( 23, 55)
( 24, 56)( 25, 57)( 26, 58)( 27, 59)( 28, 60)( 29, 61)( 30, 62)( 31, 63)
( 32, 64)( 65, 97)( 66, 98)( 67, 99)( 68,100)( 69,101)( 70,102)( 71,103)
( 72,104)( 73,105)( 74,106)( 75,107)( 76,108)( 77,109)( 78,110)( 79,111)
( 80,112)( 81,113)( 82,114)( 83,115)( 84,116)( 85,117)( 86,118)( 87,119)
( 88,120)( 89,121)( 90,122)( 91,123)( 92,124)( 93,125)( 94,126)( 95,127)
( 96,128)(129,161)(130,162)(131,163)(132,164)(133,165)(134,166)(135,167)
(136,168)(137,169)(138,170)(139,171)(140,172)(141,173)(142,174)(143,175)
(144,176)(145,177)(146,178)(147,179)(148,180)(149,181)(150,182)(151,183)
(152,184)(153,185)(154,186)(155,187)(156,188)(157,189)(158,190)(159,191)
(160,192)(193,225)(194,226)(195,227)(196,228)(197,229)(198,230)(199,231)
(200,232)(201,233)(202,234)(203,235)(204,236)(205,237)(206,238)(207,239)
(208,240)(209,241)(210,242)(211,243)(212,244)(213,245)(214,246)(215,247)
(216,248)(217,249)(218,250)(219,251)(220,252)(221,253)(222,254)(223,255)
(224,256)(257,289)(258,290)(259,291)(260,292)(261,293)(262,294)(263,295)
(264,296)(265,297)(266,298)(267,299)(268,300)(269,301)(270,302)(271,303)
(272,304)(273,305)(274,306)(275,307)(276,308)(277,309)(278,310)(279,311)
(280,312)(281,313)(282,314)(283,315)(284,316)(285,317)(286,318)(287,319)
(288,320)(321,353)(322,354)(323,355)(324,356)(325,357)(326,358)(327,359)
(328,360)(329,361)(330,362)(331,363)(332,364)(333,365)(334,366)(335,367)
(336,368)(337,369)(338,370)(339,371)(340,372)(341,373)(342,374)(343,375)
(344,376)(345,377)(346,378)(347,379)(348,380)(349,381)(350,382)(351,383)
(352,384)(385,417)(386,418)(387,419)(388,420)(389,421)(390,422)(391,423)
(392,424)(393,425)(394,426)(395,427)(396,428)(397,429)(398,430)(399,431)
(400,432)(401,433)(402,434)(403,435)(404,436)(405,437)(406,438)(407,439)
(408,440)(409,441)(410,442)(411,443)(412,444)(413,445)(414,446)(415,447)
(416,448)(449,481)(450,482)(451,483)(452,484)(453,485)(454,486)(455,487)
(456,488)(457,489)(458,490)(459,491)(460,492)(461,493)(462,494)(463,495)
(464,496)(465,497)(466,498)(467,499)(468,500)(469,501)(470,502)(471,503)
(472,504)(473,505)(474,506)(475,507)(476,508)(477,509)(478,510)(479,511)
(480,512);
s1 := Sym(512)!(  1,273)(  2,274)(  3,275)(  4,276)(  5,277)(  6,278)(  7,279)
(  8,280)(  9,281)( 10,282)( 11,283)( 12,284)( 13,285)( 14,286)( 15,287)
( 16,288)( 17,257)( 18,258)( 19,259)( 20,260)( 21,261)( 22,262)( 23,263)
( 24,264)( 25,265)( 26,266)( 27,267)( 28,268)( 29,269)( 30,270)( 31,271)
( 32,272)( 33,306)( 34,305)( 35,308)( 36,307)( 37,310)( 38,309)( 39,312)
( 40,311)( 41,314)( 42,313)( 43,316)( 44,315)( 45,318)( 46,317)( 47,320)
( 48,319)( 49,290)( 50,289)( 51,292)( 52,291)( 53,294)( 54,293)( 55,296)
( 56,295)( 57,298)( 58,297)( 59,300)( 60,299)( 61,302)( 62,301)( 63,304)
( 64,303)( 65,341)( 66,342)( 67,343)( 68,344)( 69,337)( 70,338)( 71,339)
( 72,340)( 73,349)( 74,350)( 75,351)( 76,352)( 77,345)( 78,346)( 79,347)
( 80,348)( 81,325)( 82,326)( 83,327)( 84,328)( 85,321)( 86,322)( 87,323)
( 88,324)( 89,333)( 90,334)( 91,335)( 92,336)( 93,329)( 94,330)( 95,331)
( 96,332)( 97,374)( 98,373)( 99,376)(100,375)(101,370)(102,369)(103,372)
(104,371)(105,382)(106,381)(107,384)(108,383)(109,378)(110,377)(111,380)
(112,379)(113,358)(114,357)(115,360)(116,359)(117,354)(118,353)(119,356)
(120,355)(121,366)(122,365)(123,368)(124,367)(125,362)(126,361)(127,364)
(128,363)(129,409)(130,410)(131,411)(132,412)(133,413)(134,414)(135,415)
(136,416)(137,401)(138,402)(139,403)(140,404)(141,405)(142,406)(143,407)
(144,408)(145,393)(146,394)(147,395)(148,396)(149,397)(150,398)(151,399)
(152,400)(153,385)(154,386)(155,387)(156,388)(157,389)(158,390)(159,391)
(160,392)(161,442)(162,441)(163,444)(164,443)(165,446)(166,445)(167,448)
(168,447)(169,434)(170,433)(171,436)(172,435)(173,438)(174,437)(175,440)
(176,439)(177,426)(178,425)(179,428)(180,427)(181,430)(182,429)(183,432)
(184,431)(185,418)(186,417)(187,420)(188,419)(189,422)(190,421)(191,424)
(192,423)(193,477)(194,478)(195,479)(196,480)(197,473)(198,474)(199,475)
(200,476)(201,469)(202,470)(203,471)(204,472)(205,465)(206,466)(207,467)
(208,468)(209,461)(210,462)(211,463)(212,464)(213,457)(214,458)(215,459)
(216,460)(217,453)(218,454)(219,455)(220,456)(221,449)(222,450)(223,451)
(224,452)(225,510)(226,509)(227,512)(228,511)(229,506)(230,505)(231,508)
(232,507)(233,502)(234,501)(235,504)(236,503)(237,498)(238,497)(239,500)
(240,499)(241,494)(242,493)(243,496)(244,495)(245,490)(246,489)(247,492)
(248,491)(249,486)(250,485)(251,488)(252,487)(253,482)(254,481)(255,484)
(256,483);
s2 := Sym(512)!(  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 69)(  6, 70)(  7, 71)
(  8, 72)(  9, 73)( 10, 74)( 11, 75)( 12, 76)( 13, 77)( 14, 78)( 15, 79)
( 16, 80)( 17, 81)( 18, 82)( 19, 83)( 20, 84)( 21, 85)( 22, 86)( 23, 87)
( 24, 88)( 25, 89)( 26, 90)( 27, 91)( 28, 92)( 29, 93)( 30, 94)( 31, 95)
( 32, 96)( 33, 97)( 34, 98)( 35, 99)( 36,100)( 37,101)( 38,102)( 39,103)
( 40,104)( 41,105)( 42,106)( 43,107)( 44,108)( 45,109)( 46,110)( 47,111)
( 48,112)( 49,113)( 50,114)( 51,115)( 52,116)( 53,117)( 54,118)( 55,119)
( 56,120)( 57,121)( 58,122)( 59,123)( 60,124)( 61,125)( 62,126)( 63,127)
( 64,128)(129,193)(130,194)(131,195)(132,196)(133,197)(134,198)(135,199)
(136,200)(137,201)(138,202)(139,203)(140,204)(141,205)(142,206)(143,207)
(144,208)(145,209)(146,210)(147,211)(148,212)(149,213)(150,214)(151,215)
(152,216)(153,217)(154,218)(155,219)(156,220)(157,221)(158,222)(159,223)
(160,224)(161,225)(162,226)(163,227)(164,228)(165,229)(166,230)(167,231)
(168,232)(169,233)(170,234)(171,235)(172,236)(173,237)(174,238)(175,239)
(176,240)(177,241)(178,242)(179,243)(180,244)(181,245)(182,246)(183,247)
(184,248)(185,249)(186,250)(187,251)(188,252)(189,253)(190,254)(191,255)
(192,256)(257,321)(258,322)(259,323)(260,324)(261,325)(262,326)(263,327)
(264,328)(265,329)(266,330)(267,331)(268,332)(269,333)(270,334)(271,335)
(272,336)(273,337)(274,338)(275,339)(276,340)(277,341)(278,342)(279,343)
(280,344)(281,345)(282,346)(283,347)(284,348)(285,349)(286,350)(287,351)
(288,352)(289,353)(290,354)(291,355)(292,356)(293,357)(294,358)(295,359)
(296,360)(297,361)(298,362)(299,363)(300,364)(301,365)(302,366)(303,367)
(304,368)(305,369)(306,370)(307,371)(308,372)(309,373)(310,374)(311,375)
(312,376)(313,377)(314,378)(315,379)(316,380)(317,381)(318,382)(319,383)
(320,384)(385,449)(386,450)(387,451)(388,452)(389,453)(390,454)(391,455)
(392,456)(393,457)(394,458)(395,459)(396,460)(397,461)(398,462)(399,463)
(400,464)(401,465)(402,466)(403,467)(404,468)(405,469)(406,470)(407,471)
(408,472)(409,473)(410,474)(411,475)(412,476)(413,477)(414,478)(415,479)
(416,480)(417,481)(418,482)(419,483)(420,484)(421,485)(422,486)(423,487)
(424,488)(425,489)(426,490)(427,491)(428,492)(429,493)(430,494)(431,495)
(432,496)(433,497)(434,498)(435,499)(436,500)(437,501)(438,502)(439,503)
(440,504)(441,505)(442,506)(443,507)(444,508)(445,509)(446,510)(447,511)
(448,512);
s3 := Sym(512)!(  1,129)(  2,130)(  3,131)(  4,132)(  5,133)(  6,134)(  7,135)
(  8,136)(  9,137)( 10,138)( 11,139)( 12,140)( 13,141)( 14,142)( 15,143)
( 16,144)( 17,153)( 18,154)( 19,155)( 20,156)( 21,157)( 22,158)( 23,159)
( 24,160)( 25,145)( 26,146)( 27,147)( 28,148)( 29,149)( 30,150)( 31,151)
( 32,152)( 33,161)( 34,162)( 35,163)( 36,164)( 37,165)( 38,166)( 39,167)
( 40,168)( 41,169)( 42,170)( 43,171)( 44,172)( 45,173)( 46,174)( 47,175)
( 48,176)( 49,185)( 50,186)( 51,187)( 52,188)( 53,189)( 54,190)( 55,191)
( 56,192)( 57,177)( 58,178)( 59,179)( 60,180)( 61,181)( 62,182)( 63,183)
( 64,184)( 65,195)( 66,196)( 67,193)( 68,194)( 69,199)( 70,200)( 71,197)
( 72,198)( 73,203)( 74,204)( 75,201)( 76,202)( 77,207)( 78,208)( 79,205)
( 80,206)( 81,219)( 82,220)( 83,217)( 84,218)( 85,223)( 86,224)( 87,221)
( 88,222)( 89,211)( 90,212)( 91,209)( 92,210)( 93,215)( 94,216)( 95,213)
( 96,214)( 97,227)( 98,228)( 99,225)(100,226)(101,231)(102,232)(103,229)
(104,230)(105,235)(106,236)(107,233)(108,234)(109,239)(110,240)(111,237)
(112,238)(113,251)(114,252)(115,249)(116,250)(117,255)(118,256)(119,253)
(120,254)(121,243)(122,244)(123,241)(124,242)(125,247)(126,248)(127,245)
(128,246)(257,385)(258,386)(259,387)(260,388)(261,389)(262,390)(263,391)
(264,392)(265,393)(266,394)(267,395)(268,396)(269,397)(270,398)(271,399)
(272,400)(273,409)(274,410)(275,411)(276,412)(277,413)(278,414)(279,415)
(280,416)(281,401)(282,402)(283,403)(284,404)(285,405)(286,406)(287,407)
(288,408)(289,417)(290,418)(291,419)(292,420)(293,421)(294,422)(295,423)
(296,424)(297,425)(298,426)(299,427)(300,428)(301,429)(302,430)(303,431)
(304,432)(305,441)(306,442)(307,443)(308,444)(309,445)(310,446)(311,447)
(312,448)(313,433)(314,434)(315,435)(316,436)(317,437)(318,438)(319,439)
(320,440)(321,451)(322,452)(323,449)(324,450)(325,455)(326,456)(327,453)
(328,454)(329,459)(330,460)(331,457)(332,458)(333,463)(334,464)(335,461)
(336,462)(337,475)(338,476)(339,473)(340,474)(341,479)(342,480)(343,477)
(344,478)(345,467)(346,468)(347,465)(348,466)(349,471)(350,472)(351,469)
(352,470)(353,483)(354,484)(355,481)(356,482)(357,487)(358,488)(359,485)
(360,486)(361,491)(362,492)(363,489)(364,490)(365,495)(366,496)(367,493)
(368,494)(369,507)(370,508)(371,505)(372,506)(373,511)(374,512)(375,509)
(376,510)(377,499)(378,500)(379,497)(380,498)(381,503)(382,504)(383,501)
(384,502);
s4 := Sym(512)!(  1, 17)(  2, 18)(  3, 19)(  4, 20)(  5, 21)(  6, 22)(  7, 23)
(  8, 24)(  9, 25)( 10, 26)( 11, 27)( 12, 28)( 13, 29)( 14, 30)( 15, 31)
( 16, 32)( 33, 49)( 34, 50)( 35, 51)( 36, 52)( 37, 53)( 38, 54)( 39, 55)
( 40, 56)( 41, 57)( 42, 58)( 43, 59)( 44, 60)( 45, 61)( 46, 62)( 47, 63)
( 48, 64)( 65, 81)( 66, 82)( 67, 83)( 68, 84)( 69, 85)( 70, 86)( 71, 87)
( 72, 88)( 73, 89)( 74, 90)( 75, 91)( 76, 92)( 77, 93)( 78, 94)( 79, 95)
( 80, 96)( 97,113)( 98,114)( 99,115)(100,116)(101,117)(102,118)(103,119)
(104,120)(105,121)(106,122)(107,123)(108,124)(109,125)(110,126)(111,127)
(112,128)(129,145)(130,146)(131,147)(132,148)(133,149)(134,150)(135,151)
(136,152)(137,153)(138,154)(139,155)(140,156)(141,157)(142,158)(143,159)
(144,160)(161,177)(162,178)(163,179)(164,180)(165,181)(166,182)(167,183)
(168,184)(169,185)(170,186)(171,187)(172,188)(173,189)(174,190)(175,191)
(176,192)(193,209)(194,210)(195,211)(196,212)(197,213)(198,214)(199,215)
(200,216)(201,217)(202,218)(203,219)(204,220)(205,221)(206,222)(207,223)
(208,224)(225,241)(226,242)(227,243)(228,244)(229,245)(230,246)(231,247)
(232,248)(233,249)(234,250)(235,251)(236,252)(237,253)(238,254)(239,255)
(240,256)(257,273)(258,274)(259,275)(260,276)(261,277)(262,278)(263,279)
(264,280)(265,281)(266,282)(267,283)(268,284)(269,285)(270,286)(271,287)
(272,288)(289,305)(290,306)(291,307)(292,308)(293,309)(294,310)(295,311)
(296,312)(297,313)(298,314)(299,315)(300,316)(301,317)(302,318)(303,319)
(304,320)(321,337)(322,338)(323,339)(324,340)(325,341)(326,342)(327,343)
(328,344)(329,345)(330,346)(331,347)(332,348)(333,349)(334,350)(335,351)
(336,352)(353,369)(354,370)(355,371)(356,372)(357,373)(358,374)(359,375)
(360,376)(361,377)(362,378)(363,379)(364,380)(365,381)(366,382)(367,383)
(368,384)(385,401)(386,402)(387,403)(388,404)(389,405)(390,406)(391,407)
(392,408)(393,409)(394,410)(395,411)(396,412)(397,413)(398,414)(399,415)
(400,416)(417,433)(418,434)(419,435)(420,436)(421,437)(422,438)(423,439)
(424,440)(425,441)(426,442)(427,443)(428,444)(429,445)(430,446)(431,447)
(432,448)(449,465)(450,466)(451,467)(452,468)(453,469)(454,470)(455,471)
(456,472)(457,473)(458,474)(459,475)(460,476)(461,477)(462,478)(463,479)
(464,480)(481,497)(482,498)(483,499)(484,500)(485,501)(486,502)(487,503)
(488,504)(489,505)(490,506)(491,507)(492,508)(493,509)(494,510)(495,511)
(496,512);
poly := sub<Sym(512)|s0,s1,s2,s3,s4>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4 >; 
 
References : None.
to this polytope