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Polytope of Type {4,4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,4}*512b
Also Known As : 2T4(4,0)(2,0), {{4,4|4},{4,4|2}}. if this polytope has another name.
Group : SmallGroup(512,406983)
Rank : 4
Schlafli Type : {4,4,4}
Number of vertices, edges, etc : 16, 32, 32, 4
Order of s0s1s2s3 : 8
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Locally Toroidal
   Orientable
   Flat
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 : {4,4,4}*256b, {4,4,2}*256
   4-fold quotients : {4,4,4}*128, {4,4,2}*128
   8-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64
   16-fold quotients : {2,2,4}*32, {2,4,2}*32, {4,2,2}*32
   32-fold quotients : {2,2,2}*16
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, 44)( 10, 43)( 11, 42)( 12, 41)( 13, 48)( 14, 47)( 15, 46)( 16, 45)
( 17, 49)( 18, 50)( 19, 51)( 20, 52)( 21, 53)( 22, 54)( 23, 55)( 24, 56)
( 25, 60)( 26, 59)( 27, 58)( 28, 57)( 29, 64)( 30, 63)( 31, 62)( 32, 61)
( 65, 97)( 66, 98)( 67, 99)( 68,100)( 69,101)( 70,102)( 71,103)( 72,104)
( 73,108)( 74,107)( 75,106)( 76,105)( 77,112)( 78,111)( 79,110)( 80,109)
( 81,113)( 82,114)( 83,115)( 84,116)( 85,117)( 86,118)( 87,119)( 88,120)
( 89,124)( 90,123)( 91,122)( 92,121)( 93,128)( 94,127)( 95,126)( 96,125)
(129,161)(130,162)(131,163)(132,164)(133,165)(134,166)(135,167)(136,168)
(137,172)(138,171)(139,170)(140,169)(141,176)(142,175)(143,174)(144,173)
(145,177)(146,178)(147,179)(148,180)(149,181)(150,182)(151,183)(152,184)
(153,188)(154,187)(155,186)(156,185)(157,192)(158,191)(159,190)(160,189)
(193,225)(194,226)(195,227)(196,228)(197,229)(198,230)(199,231)(200,232)
(201,236)(202,235)(203,234)(204,233)(205,240)(206,239)(207,238)(208,237)
(209,241)(210,242)(211,243)(212,244)(213,245)(214,246)(215,247)(216,248)
(217,252)(218,251)(219,250)(220,249)(221,256)(222,255)(223,254)(224,253)
(257,289)(258,290)(259,291)(260,292)(261,293)(262,294)(263,295)(264,296)
(265,300)(266,299)(267,298)(268,297)(269,304)(270,303)(271,302)(272,301)
(273,305)(274,306)(275,307)(276,308)(277,309)(278,310)(279,311)(280,312)
(281,316)(282,315)(283,314)(284,313)(285,320)(286,319)(287,318)(288,317)
(321,353)(322,354)(323,355)(324,356)(325,357)(326,358)(327,359)(328,360)
(329,364)(330,363)(331,362)(332,361)(333,368)(334,367)(335,366)(336,365)
(337,369)(338,370)(339,371)(340,372)(341,373)(342,374)(343,375)(344,376)
(345,380)(346,379)(347,378)(348,377)(349,384)(350,383)(351,382)(352,381)
(385,417)(386,418)(387,419)(388,420)(389,421)(390,422)(391,423)(392,424)
(393,428)(394,427)(395,426)(396,425)(397,432)(398,431)(399,430)(400,429)
(401,433)(402,434)(403,435)(404,436)(405,437)(406,438)(407,439)(408,440)
(409,444)(410,443)(411,442)(412,441)(413,448)(414,447)(415,446)(416,445)
(449,481)(450,482)(451,483)(452,484)(453,485)(454,486)(455,487)(456,488)
(457,492)(458,491)(459,490)(460,489)(461,496)(462,495)(463,494)(464,493)
(465,497)(466,498)(467,499)(468,500)(469,501)(470,502)(471,503)(472,504)
(473,508)(474,507)(475,506)(476,505)(477,512)(478,511)(479,510)(480,509);;
s1 := (  1,257)(  2,258)(  3,260)(  4,259)(  5,262)(  6,261)(  7,263)(  8,264)
(  9,265)( 10,266)( 11,268)( 12,267)( 13,270)( 14,269)( 15,271)( 16,272)
( 17,273)( 18,274)( 19,276)( 20,275)( 21,278)( 22,277)( 23,279)( 24,280)
( 25,281)( 26,282)( 27,284)( 28,283)( 29,286)( 30,285)( 31,287)( 32,288)
( 33,305)( 34,306)( 35,308)( 36,307)( 37,310)( 38,309)( 39,311)( 40,312)
( 41,313)( 42,314)( 43,316)( 44,315)( 45,318)( 46,317)( 47,319)( 48,320)
( 49,289)( 50,290)( 51,292)( 52,291)( 53,294)( 54,293)( 55,295)( 56,296)
( 57,297)( 58,298)( 59,300)( 60,299)( 61,302)( 62,301)( 63,303)( 64,304)
( 65,329)( 66,330)( 67,332)( 68,331)( 69,334)( 70,333)( 71,335)( 72,336)
( 73,321)( 74,322)( 75,324)( 76,323)( 77,326)( 78,325)( 79,327)( 80,328)
( 81,346)( 82,345)( 83,347)( 84,348)( 85,349)( 86,350)( 87,352)( 88,351)
( 89,338)( 90,337)( 91,339)( 92,340)( 93,341)( 94,342)( 95,344)( 96,343)
( 97,379)( 98,380)( 99,378)(100,377)(101,384)(102,383)(103,381)(104,382)
(105,371)(106,372)(107,370)(108,369)(109,376)(110,375)(111,373)(112,374)
(113,364)(114,363)(115,361)(116,362)(117,367)(118,368)(119,366)(120,365)
(121,356)(122,355)(123,353)(124,354)(125,359)(126,360)(127,358)(128,357)
(129,401)(130,402)(131,404)(132,403)(133,406)(134,405)(135,407)(136,408)
(137,409)(138,410)(139,412)(140,411)(141,414)(142,413)(143,415)(144,416)
(145,385)(146,386)(147,388)(148,387)(149,390)(150,389)(151,391)(152,392)
(153,393)(154,394)(155,396)(156,395)(157,398)(158,397)(159,399)(160,400)
(161,417)(162,418)(163,420)(164,419)(165,422)(166,421)(167,423)(168,424)
(169,425)(170,426)(171,428)(172,427)(173,430)(174,429)(175,431)(176,432)
(177,433)(178,434)(179,436)(180,435)(181,438)(182,437)(183,439)(184,440)
(185,441)(186,442)(187,444)(188,443)(189,446)(190,445)(191,447)(192,448)
(193,475)(194,476)(195,474)(196,473)(197,480)(198,479)(199,477)(200,478)
(201,467)(202,468)(203,466)(204,465)(205,472)(206,471)(207,469)(208,470)
(209,460)(210,459)(211,457)(212,458)(213,463)(214,464)(215,462)(216,461)
(217,452)(218,451)(219,449)(220,450)(221,455)(222,456)(223,454)(224,453)
(225,489)(226,490)(227,492)(228,491)(229,494)(230,493)(231,495)(232,496)
(233,481)(234,482)(235,484)(236,483)(237,486)(238,485)(239,487)(240,488)
(241,506)(242,505)(243,507)(244,508)(245,509)(246,510)(247,512)(248,511)
(249,498)(250,497)(251,499)(252,500)(253,501)(254,502)(255,504)(256,503);;
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, 83)( 18, 84)( 19, 81)( 20, 82)( 21, 87)( 22, 88)( 23, 85)( 24, 86)
( 25, 91)( 26, 92)( 27, 89)( 28, 90)( 29, 95)( 30, 96)( 31, 93)( 32, 94)
( 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)
( 49,115)( 50,116)( 51,113)( 52,114)( 53,119)( 54,120)( 55,117)( 56,118)
( 57,123)( 58,124)( 59,121)( 60,122)( 61,127)( 62,128)( 63,125)( 64,126)
(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)
(145,211)(146,212)(147,209)(148,210)(149,215)(150,216)(151,213)(152,214)
(153,219)(154,220)(155,217)(156,218)(157,223)(158,224)(159,221)(160,222)
(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)
(177,243)(178,244)(179,241)(180,242)(181,247)(182,248)(183,245)(184,246)
(185,251)(186,252)(187,249)(188,250)(189,255)(190,256)(191,253)(192,254)
(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)
(273,339)(274,340)(275,337)(276,338)(277,343)(278,344)(279,341)(280,342)
(281,347)(282,348)(283,345)(284,346)(285,351)(286,352)(287,349)(288,350)
(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)
(305,371)(306,372)(307,369)(308,370)(309,375)(310,376)(311,373)(312,374)
(313,379)(314,380)(315,377)(316,378)(317,383)(318,384)(319,381)(320,382)
(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)
(401,467)(402,468)(403,465)(404,466)(405,471)(406,472)(407,469)(408,470)
(409,475)(410,476)(411,473)(412,474)(413,479)(414,480)(415,477)(416,478)
(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)
(433,499)(434,500)(435,497)(436,498)(437,503)(438,504)(439,501)(440,502)
(441,507)(442,508)(443,505)(444,506)(445,511)(446,512)(447,509)(448,510);;
s3 := (  1,177)(  2,178)(  3,179)(  4,180)(  5,181)(  6,182)(  7,183)(  8,184)
(  9,185)( 10,186)( 11,187)( 12,188)( 13,189)( 14,190)( 15,191)( 16,192)
( 17,161)( 18,162)( 19,163)( 20,164)( 21,165)( 22,166)( 23,167)( 24,168)
( 25,169)( 26,170)( 27,171)( 28,172)( 29,173)( 30,174)( 31,175)( 32,176)
( 33,145)( 34,146)( 35,147)( 36,148)( 37,149)( 38,150)( 39,151)( 40,152)
( 41,153)( 42,154)( 43,155)( 44,156)( 45,157)( 46,158)( 47,159)( 48,160)
( 49,129)( 50,130)( 51,131)( 52,132)( 53,133)( 54,134)( 55,135)( 56,136)
( 57,137)( 58,138)( 59,139)( 60,140)( 61,141)( 62,142)( 63,143)( 64,144)
( 65,245)( 66,246)( 67,247)( 68,248)( 69,241)( 70,242)( 71,243)( 72,244)
( 73,253)( 74,254)( 75,255)( 76,256)( 77,249)( 78,250)( 79,251)( 80,252)
( 81,229)( 82,230)( 83,231)( 84,232)( 85,225)( 86,226)( 87,227)( 88,228)
( 89,237)( 90,238)( 91,239)( 92,240)( 93,233)( 94,234)( 95,235)( 96,236)
( 97,213)( 98,214)( 99,215)(100,216)(101,209)(102,210)(103,211)(104,212)
(105,221)(106,222)(107,223)(108,224)(109,217)(110,218)(111,219)(112,220)
(113,197)(114,198)(115,199)(116,200)(117,193)(118,194)(119,195)(120,196)
(121,205)(122,206)(123,207)(124,208)(125,201)(126,202)(127,203)(128,204)
(257,433)(258,434)(259,435)(260,436)(261,437)(262,438)(263,439)(264,440)
(265,441)(266,442)(267,443)(268,444)(269,445)(270,446)(271,447)(272,448)
(273,417)(274,418)(275,419)(276,420)(277,421)(278,422)(279,423)(280,424)
(281,425)(282,426)(283,427)(284,428)(285,429)(286,430)(287,431)(288,432)
(289,401)(290,402)(291,403)(292,404)(293,405)(294,406)(295,407)(296,408)
(297,409)(298,410)(299,411)(300,412)(301,413)(302,414)(303,415)(304,416)
(305,385)(306,386)(307,387)(308,388)(309,389)(310,390)(311,391)(312,392)
(313,393)(314,394)(315,395)(316,396)(317,397)(318,398)(319,399)(320,400)
(321,501)(322,502)(323,503)(324,504)(325,497)(326,498)(327,499)(328,500)
(329,509)(330,510)(331,511)(332,512)(333,505)(334,506)(335,507)(336,508)
(337,485)(338,486)(339,487)(340,488)(341,481)(342,482)(343,483)(344,484)
(345,493)(346,494)(347,495)(348,496)(349,489)(350,490)(351,491)(352,492)
(353,469)(354,470)(355,471)(356,472)(357,465)(358,466)(359,467)(360,468)
(361,477)(362,478)(363,479)(364,480)(365,473)(366,474)(367,475)(368,476)
(369,453)(370,454)(371,455)(372,456)(373,449)(374,450)(375,451)(376,452)
(377,461)(378,462)(379,463)(380,464)(381,457)(382,458)(383,459)(384,460);;
poly := Group([s0,s1,s2,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*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, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
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, 44)( 10, 43)( 11, 42)( 12, 41)( 13, 48)( 14, 47)( 15, 46)
( 16, 45)( 17, 49)( 18, 50)( 19, 51)( 20, 52)( 21, 53)( 22, 54)( 23, 55)
( 24, 56)( 25, 60)( 26, 59)( 27, 58)( 28, 57)( 29, 64)( 30, 63)( 31, 62)
( 32, 61)( 65, 97)( 66, 98)( 67, 99)( 68,100)( 69,101)( 70,102)( 71,103)
( 72,104)( 73,108)( 74,107)( 75,106)( 76,105)( 77,112)( 78,111)( 79,110)
( 80,109)( 81,113)( 82,114)( 83,115)( 84,116)( 85,117)( 86,118)( 87,119)
( 88,120)( 89,124)( 90,123)( 91,122)( 92,121)( 93,128)( 94,127)( 95,126)
( 96,125)(129,161)(130,162)(131,163)(132,164)(133,165)(134,166)(135,167)
(136,168)(137,172)(138,171)(139,170)(140,169)(141,176)(142,175)(143,174)
(144,173)(145,177)(146,178)(147,179)(148,180)(149,181)(150,182)(151,183)
(152,184)(153,188)(154,187)(155,186)(156,185)(157,192)(158,191)(159,190)
(160,189)(193,225)(194,226)(195,227)(196,228)(197,229)(198,230)(199,231)
(200,232)(201,236)(202,235)(203,234)(204,233)(205,240)(206,239)(207,238)
(208,237)(209,241)(210,242)(211,243)(212,244)(213,245)(214,246)(215,247)
(216,248)(217,252)(218,251)(219,250)(220,249)(221,256)(222,255)(223,254)
(224,253)(257,289)(258,290)(259,291)(260,292)(261,293)(262,294)(263,295)
(264,296)(265,300)(266,299)(267,298)(268,297)(269,304)(270,303)(271,302)
(272,301)(273,305)(274,306)(275,307)(276,308)(277,309)(278,310)(279,311)
(280,312)(281,316)(282,315)(283,314)(284,313)(285,320)(286,319)(287,318)
(288,317)(321,353)(322,354)(323,355)(324,356)(325,357)(326,358)(327,359)
(328,360)(329,364)(330,363)(331,362)(332,361)(333,368)(334,367)(335,366)
(336,365)(337,369)(338,370)(339,371)(340,372)(341,373)(342,374)(343,375)
(344,376)(345,380)(346,379)(347,378)(348,377)(349,384)(350,383)(351,382)
(352,381)(385,417)(386,418)(387,419)(388,420)(389,421)(390,422)(391,423)
(392,424)(393,428)(394,427)(395,426)(396,425)(397,432)(398,431)(399,430)
(400,429)(401,433)(402,434)(403,435)(404,436)(405,437)(406,438)(407,439)
(408,440)(409,444)(410,443)(411,442)(412,441)(413,448)(414,447)(415,446)
(416,445)(449,481)(450,482)(451,483)(452,484)(453,485)(454,486)(455,487)
(456,488)(457,492)(458,491)(459,490)(460,489)(461,496)(462,495)(463,494)
(464,493)(465,497)(466,498)(467,499)(468,500)(469,501)(470,502)(471,503)
(472,504)(473,508)(474,507)(475,506)(476,505)(477,512)(478,511)(479,510)
(480,509);
s1 := Sym(512)!(  1,257)(  2,258)(  3,260)(  4,259)(  5,262)(  6,261)(  7,263)
(  8,264)(  9,265)( 10,266)( 11,268)( 12,267)( 13,270)( 14,269)( 15,271)
( 16,272)( 17,273)( 18,274)( 19,276)( 20,275)( 21,278)( 22,277)( 23,279)
( 24,280)( 25,281)( 26,282)( 27,284)( 28,283)( 29,286)( 30,285)( 31,287)
( 32,288)( 33,305)( 34,306)( 35,308)( 36,307)( 37,310)( 38,309)( 39,311)
( 40,312)( 41,313)( 42,314)( 43,316)( 44,315)( 45,318)( 46,317)( 47,319)
( 48,320)( 49,289)( 50,290)( 51,292)( 52,291)( 53,294)( 54,293)( 55,295)
( 56,296)( 57,297)( 58,298)( 59,300)( 60,299)( 61,302)( 62,301)( 63,303)
( 64,304)( 65,329)( 66,330)( 67,332)( 68,331)( 69,334)( 70,333)( 71,335)
( 72,336)( 73,321)( 74,322)( 75,324)( 76,323)( 77,326)( 78,325)( 79,327)
( 80,328)( 81,346)( 82,345)( 83,347)( 84,348)( 85,349)( 86,350)( 87,352)
( 88,351)( 89,338)( 90,337)( 91,339)( 92,340)( 93,341)( 94,342)( 95,344)
( 96,343)( 97,379)( 98,380)( 99,378)(100,377)(101,384)(102,383)(103,381)
(104,382)(105,371)(106,372)(107,370)(108,369)(109,376)(110,375)(111,373)
(112,374)(113,364)(114,363)(115,361)(116,362)(117,367)(118,368)(119,366)
(120,365)(121,356)(122,355)(123,353)(124,354)(125,359)(126,360)(127,358)
(128,357)(129,401)(130,402)(131,404)(132,403)(133,406)(134,405)(135,407)
(136,408)(137,409)(138,410)(139,412)(140,411)(141,414)(142,413)(143,415)
(144,416)(145,385)(146,386)(147,388)(148,387)(149,390)(150,389)(151,391)
(152,392)(153,393)(154,394)(155,396)(156,395)(157,398)(158,397)(159,399)
(160,400)(161,417)(162,418)(163,420)(164,419)(165,422)(166,421)(167,423)
(168,424)(169,425)(170,426)(171,428)(172,427)(173,430)(174,429)(175,431)
(176,432)(177,433)(178,434)(179,436)(180,435)(181,438)(182,437)(183,439)
(184,440)(185,441)(186,442)(187,444)(188,443)(189,446)(190,445)(191,447)
(192,448)(193,475)(194,476)(195,474)(196,473)(197,480)(198,479)(199,477)
(200,478)(201,467)(202,468)(203,466)(204,465)(205,472)(206,471)(207,469)
(208,470)(209,460)(210,459)(211,457)(212,458)(213,463)(214,464)(215,462)
(216,461)(217,452)(218,451)(219,449)(220,450)(221,455)(222,456)(223,454)
(224,453)(225,489)(226,490)(227,492)(228,491)(229,494)(230,493)(231,495)
(232,496)(233,481)(234,482)(235,484)(236,483)(237,486)(238,485)(239,487)
(240,488)(241,506)(242,505)(243,507)(244,508)(245,509)(246,510)(247,512)
(248,511)(249,498)(250,497)(251,499)(252,500)(253,501)(254,502)(255,504)
(256,503);
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, 83)( 18, 84)( 19, 81)( 20, 82)( 21, 87)( 22, 88)( 23, 85)
( 24, 86)( 25, 91)( 26, 92)( 27, 89)( 28, 90)( 29, 95)( 30, 96)( 31, 93)
( 32, 94)( 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)( 49,115)( 50,116)( 51,113)( 52,114)( 53,119)( 54,120)( 55,117)
( 56,118)( 57,123)( 58,124)( 59,121)( 60,122)( 61,127)( 62,128)( 63,125)
( 64,126)(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)(145,211)(146,212)(147,209)(148,210)(149,215)(150,216)(151,213)
(152,214)(153,219)(154,220)(155,217)(156,218)(157,223)(158,224)(159,221)
(160,222)(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)(177,243)(178,244)(179,241)(180,242)(181,247)(182,248)(183,245)
(184,246)(185,251)(186,252)(187,249)(188,250)(189,255)(190,256)(191,253)
(192,254)(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)(273,339)(274,340)(275,337)(276,338)(277,343)(278,344)(279,341)
(280,342)(281,347)(282,348)(283,345)(284,346)(285,351)(286,352)(287,349)
(288,350)(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)(305,371)(306,372)(307,369)(308,370)(309,375)(310,376)(311,373)
(312,374)(313,379)(314,380)(315,377)(316,378)(317,383)(318,384)(319,381)
(320,382)(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)(401,467)(402,468)(403,465)(404,466)(405,471)(406,472)(407,469)
(408,470)(409,475)(410,476)(411,473)(412,474)(413,479)(414,480)(415,477)
(416,478)(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)(433,499)(434,500)(435,497)(436,498)(437,503)(438,504)(439,501)
(440,502)(441,507)(442,508)(443,505)(444,506)(445,511)(446,512)(447,509)
(448,510);
s3 := Sym(512)!(  1,177)(  2,178)(  3,179)(  4,180)(  5,181)(  6,182)(  7,183)
(  8,184)(  9,185)( 10,186)( 11,187)( 12,188)( 13,189)( 14,190)( 15,191)
( 16,192)( 17,161)( 18,162)( 19,163)( 20,164)( 21,165)( 22,166)( 23,167)
( 24,168)( 25,169)( 26,170)( 27,171)( 28,172)( 29,173)( 30,174)( 31,175)
( 32,176)( 33,145)( 34,146)( 35,147)( 36,148)( 37,149)( 38,150)( 39,151)
( 40,152)( 41,153)( 42,154)( 43,155)( 44,156)( 45,157)( 46,158)( 47,159)
( 48,160)( 49,129)( 50,130)( 51,131)( 52,132)( 53,133)( 54,134)( 55,135)
( 56,136)( 57,137)( 58,138)( 59,139)( 60,140)( 61,141)( 62,142)( 63,143)
( 64,144)( 65,245)( 66,246)( 67,247)( 68,248)( 69,241)( 70,242)( 71,243)
( 72,244)( 73,253)( 74,254)( 75,255)( 76,256)( 77,249)( 78,250)( 79,251)
( 80,252)( 81,229)( 82,230)( 83,231)( 84,232)( 85,225)( 86,226)( 87,227)
( 88,228)( 89,237)( 90,238)( 91,239)( 92,240)( 93,233)( 94,234)( 95,235)
( 96,236)( 97,213)( 98,214)( 99,215)(100,216)(101,209)(102,210)(103,211)
(104,212)(105,221)(106,222)(107,223)(108,224)(109,217)(110,218)(111,219)
(112,220)(113,197)(114,198)(115,199)(116,200)(117,193)(118,194)(119,195)
(120,196)(121,205)(122,206)(123,207)(124,208)(125,201)(126,202)(127,203)
(128,204)(257,433)(258,434)(259,435)(260,436)(261,437)(262,438)(263,439)
(264,440)(265,441)(266,442)(267,443)(268,444)(269,445)(270,446)(271,447)
(272,448)(273,417)(274,418)(275,419)(276,420)(277,421)(278,422)(279,423)
(280,424)(281,425)(282,426)(283,427)(284,428)(285,429)(286,430)(287,431)
(288,432)(289,401)(290,402)(291,403)(292,404)(293,405)(294,406)(295,407)
(296,408)(297,409)(298,410)(299,411)(300,412)(301,413)(302,414)(303,415)
(304,416)(305,385)(306,386)(307,387)(308,388)(309,389)(310,390)(311,391)
(312,392)(313,393)(314,394)(315,395)(316,396)(317,397)(318,398)(319,399)
(320,400)(321,501)(322,502)(323,503)(324,504)(325,497)(326,498)(327,499)
(328,500)(329,509)(330,510)(331,511)(332,512)(333,505)(334,506)(335,507)
(336,508)(337,485)(338,486)(339,487)(340,488)(341,481)(342,482)(343,483)
(344,484)(345,493)(346,494)(347,495)(348,496)(349,489)(350,490)(351,491)
(352,492)(353,469)(354,470)(355,471)(356,472)(357,465)(358,466)(359,467)
(360,468)(361,477)(362,478)(363,479)(364,480)(365,473)(366,474)(367,475)
(368,476)(369,453)(370,454)(371,455)(372,456)(373,449)(374,450)(375,451)
(376,452)(377,461)(378,462)(379,463)(380,464)(381,457)(382,458)(383,459)
(384,460);
poly := sub<Sym(512)|s0,s1,s2,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*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, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >; 
 
References :
  1. Theorem 10C2, McMullen P., Schulte, E.; Abstract Regular Polytopes (Cambr\ idge University Press, 2002)

to this polytope