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# Polytope of Type {8,4,4}

Atlas Canonical Name : {8,4,4}*512b
Also Known As : {{8,4|2},{4,4}4}. if this polytope has another name.
Group : SmallGroup(512,155651)
Rank : 4
Schlafli Type : {8,4,4}
Number of vertices, edges, etc : 8, 32, 16, 8
Order of s0s1s2s3 : 8
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
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 : {8,4,4}*256a, {4,4,4}*256a
4-fold quotients : {4,4,4}*128, {8,4,2}*128a, {2,4,4}*128, {8,2,4}*128
8-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64, {8,2,2}*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, 41)( 10, 42)( 11, 43)( 12, 44)( 13, 45)( 14, 46)( 15, 47)( 16, 48)
( 17, 51)( 18, 52)( 19, 49)( 20, 50)( 21, 55)( 22, 56)( 23, 53)( 24, 54)
( 25, 59)( 26, 60)( 27, 57)( 28, 58)( 29, 63)( 30, 64)( 31, 61)( 32, 62)
( 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)
( 81,115)( 82,116)( 83,113)( 84,114)( 85,119)( 86,120)( 87,117)( 88,118)
( 89,123)( 90,124)( 91,121)( 92,122)( 93,127)( 94,128)( 95,125)( 96,126)
(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)
(145,179)(146,180)(147,177)(148,178)(149,183)(150,184)(151,181)(152,182)
(153,187)(154,188)(155,185)(156,186)(157,191)(158,192)(159,189)(160,190)
(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)
(209,243)(210,244)(211,241)(212,242)(213,247)(214,248)(215,245)(216,246)
(217,251)(218,252)(219,249)(220,250)(221,255)(222,256)(223,253)(224,254)
(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)
(273,307)(274,308)(275,305)(276,306)(277,311)(278,312)(279,309)(280,310)
(281,315)(282,316)(283,313)(284,314)(285,319)(286,320)(287,317)(288,318)
(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)
(337,371)(338,372)(339,369)(340,370)(341,375)(342,376)(343,373)(344,374)
(345,379)(346,380)(347,377)(348,378)(349,383)(350,384)(351,381)(352,382)
(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)
(401,435)(402,436)(403,433)(404,434)(405,439)(406,440)(407,437)(408,438)
(409,443)(410,444)(411,441)(412,442)(413,447)(414,448)(415,445)(416,446)
(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)
(465,499)(466,500)(467,497)(468,498)(469,503)(470,504)(471,501)(472,502)
(473,507)(474,508)(475,505)(476,506)(477,511)(478,512)(479,509)(480,510);;
s1 := (  1,257)(  2,258)(  3,259)(  4,260)(  5,262)(  6,261)(  7,264)(  8,263)
(  9,265)( 10,266)( 11,267)( 12,268)( 13,270)( 14,269)( 15,272)( 16,271)
( 17,275)( 18,276)( 19,273)( 20,274)( 21,280)( 22,279)( 23,278)( 24,277)
( 25,283)( 26,284)( 27,281)( 28,282)( 29,288)( 30,287)( 31,286)( 32,285)
( 33,305)( 34,306)( 35,307)( 36,308)( 37,310)( 38,309)( 39,312)( 40,311)
( 41,313)( 42,314)( 43,315)( 44,316)( 45,318)( 46,317)( 47,320)( 48,319)
( 49,289)( 50,290)( 51,291)( 52,292)( 53,294)( 54,293)( 55,296)( 56,295)
( 57,297)( 58,298)( 59,299)( 60,300)( 61,302)( 62,301)( 63,304)( 64,303)
( 65,329)( 66,330)( 67,331)( 68,332)( 69,334)( 70,333)( 71,336)( 72,335)
( 73,321)( 74,322)( 75,323)( 76,324)( 77,326)( 78,325)( 79,328)( 80,327)
( 81,347)( 82,348)( 83,345)( 84,346)( 85,352)( 86,351)( 87,350)( 88,349)
( 89,339)( 90,340)( 91,337)( 92,338)( 93,344)( 94,343)( 95,342)( 96,341)
( 97,377)( 98,378)( 99,379)(100,380)(101,382)(102,381)(103,384)(104,383)
(105,369)(106,370)(107,371)(108,372)(109,374)(110,373)(111,376)(112,375)
(113,361)(114,362)(115,363)(116,364)(117,366)(118,365)(119,368)(120,367)
(121,353)(122,354)(123,355)(124,356)(125,358)(126,357)(127,360)(128,359)
(129,401)(130,402)(131,403)(132,404)(133,406)(134,405)(135,408)(136,407)
(137,409)(138,410)(139,411)(140,412)(141,414)(142,413)(143,416)(144,415)
(145,385)(146,386)(147,387)(148,388)(149,390)(150,389)(151,392)(152,391)
(153,393)(154,394)(155,395)(156,396)(157,398)(158,397)(159,400)(160,399)
(161,417)(162,418)(163,419)(164,420)(165,422)(166,421)(167,424)(168,423)
(169,425)(170,426)(171,427)(172,428)(173,430)(174,429)(175,432)(176,431)
(177,435)(178,436)(179,433)(180,434)(181,440)(182,439)(183,438)(184,437)
(185,443)(186,444)(187,441)(188,442)(189,448)(190,447)(191,446)(192,445)
(193,473)(194,474)(195,475)(196,476)(197,478)(198,477)(199,480)(200,479)
(201,465)(202,466)(203,467)(204,468)(205,470)(206,469)(207,472)(208,471)
(209,457)(210,458)(211,459)(212,460)(213,462)(214,461)(215,464)(216,463)
(217,449)(218,450)(219,451)(220,452)(221,454)(222,453)(223,456)(224,455)
(225,489)(226,490)(227,491)(228,492)(229,494)(230,493)(231,496)(232,495)
(233,481)(234,482)(235,483)(236,484)(237,486)(238,485)(239,488)(240,487)
(241,507)(242,508)(243,505)(244,506)(245,512)(246,511)(247,510)(248,509)
(249,499)(250,500)(251,497)(252,498)(253,504)(254,503)(255,502)(256,501);;
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 := (  1,161)(  2,162)(  3,163)(  4,164)(  5,165)(  6,166)(  7,167)(  8,168)
(  9,170)( 10,169)( 11,172)( 12,171)( 13,174)( 14,173)( 15,176)( 16,175)
( 17,177)( 18,178)( 19,179)( 20,180)( 21,181)( 22,182)( 23,183)( 24,184)
( 25,186)( 26,185)( 27,188)( 28,187)( 29,190)( 30,189)( 31,192)( 32,191)
( 33,129)( 34,130)( 35,131)( 36,132)( 37,133)( 38,134)( 39,135)( 40,136)
( 41,138)( 42,137)( 43,140)( 44,139)( 45,142)( 46,141)( 47,144)( 48,143)
( 49,145)( 50,146)( 51,147)( 52,148)( 53,149)( 54,150)( 55,151)( 56,152)
( 57,154)( 58,153)( 59,156)( 60,155)( 61,158)( 62,157)( 63,160)( 64,159)
( 65,229)( 66,230)( 67,231)( 68,232)( 69,225)( 70,226)( 71,227)( 72,228)
( 73,238)( 74,237)( 75,240)( 76,239)( 77,234)( 78,233)( 79,236)( 80,235)
( 81,245)( 82,246)( 83,247)( 84,248)( 85,241)( 86,242)( 87,243)( 88,244)
( 89,254)( 90,253)( 91,256)( 92,255)( 93,250)( 94,249)( 95,252)( 96,251)
( 97,197)( 98,198)( 99,199)(100,200)(101,193)(102,194)(103,195)(104,196)
(105,206)(106,205)(107,208)(108,207)(109,202)(110,201)(111,204)(112,203)
(113,213)(114,214)(115,215)(116,216)(117,209)(118,210)(119,211)(120,212)
(121,222)(122,221)(123,224)(124,223)(125,218)(126,217)(127,220)(128,219)
(257,417)(258,418)(259,419)(260,420)(261,421)(262,422)(263,423)(264,424)
(265,426)(266,425)(267,428)(268,427)(269,430)(270,429)(271,432)(272,431)
(273,433)(274,434)(275,435)(276,436)(277,437)(278,438)(279,439)(280,440)
(281,442)(282,441)(283,444)(284,443)(285,446)(286,445)(287,448)(288,447)
(289,385)(290,386)(291,387)(292,388)(293,389)(294,390)(295,391)(296,392)
(297,394)(298,393)(299,396)(300,395)(301,398)(302,397)(303,400)(304,399)
(305,401)(306,402)(307,403)(308,404)(309,405)(310,406)(311,407)(312,408)
(313,410)(314,409)(315,412)(316,411)(317,414)(318,413)(319,416)(320,415)
(321,485)(322,486)(323,487)(324,488)(325,481)(326,482)(327,483)(328,484)
(329,494)(330,493)(331,496)(332,495)(333,490)(334,489)(335,492)(336,491)
(337,501)(338,502)(339,503)(340,504)(341,497)(342,498)(343,499)(344,500)
(345,510)(346,509)(347,512)(348,511)(349,506)(350,505)(351,508)(352,507)
(353,453)(354,454)(355,455)(356,456)(357,449)(358,450)(359,451)(360,452)
(361,462)(362,461)(363,464)(364,463)(365,458)(366,457)(367,460)(368,459)
(369,469)(370,470)(371,471)(372,472)(373,465)(374,466)(375,467)(376,468)
(377,478)(378,477)(379,480)(380,479)(381,474)(382,473)(383,476)(384,475);;
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*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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, 41)( 10, 42)( 11, 43)( 12, 44)( 13, 45)( 14, 46)( 15, 47)
( 16, 48)( 17, 51)( 18, 52)( 19, 49)( 20, 50)( 21, 55)( 22, 56)( 23, 53)
( 24, 54)( 25, 59)( 26, 60)( 27, 57)( 28, 58)( 29, 63)( 30, 64)( 31, 61)
( 32, 62)( 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)( 81,115)( 82,116)( 83,113)( 84,114)( 85,119)( 86,120)( 87,117)
( 88,118)( 89,123)( 90,124)( 91,121)( 92,122)( 93,127)( 94,128)( 95,125)
( 96,126)(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)(145,179)(146,180)(147,177)(148,178)(149,183)(150,184)(151,181)
(152,182)(153,187)(154,188)(155,185)(156,186)(157,191)(158,192)(159,189)
(160,190)(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)(209,243)(210,244)(211,241)(212,242)(213,247)(214,248)(215,245)
(216,246)(217,251)(218,252)(219,249)(220,250)(221,255)(222,256)(223,253)
(224,254)(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)(273,307)(274,308)(275,305)(276,306)(277,311)(278,312)(279,309)
(280,310)(281,315)(282,316)(283,313)(284,314)(285,319)(286,320)(287,317)
(288,318)(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)(337,371)(338,372)(339,369)(340,370)(341,375)(342,376)(343,373)
(344,374)(345,379)(346,380)(347,377)(348,378)(349,383)(350,384)(351,381)
(352,382)(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)(401,435)(402,436)(403,433)(404,434)(405,439)(406,440)(407,437)
(408,438)(409,443)(410,444)(411,441)(412,442)(413,447)(414,448)(415,445)
(416,446)(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)(465,499)(466,500)(467,497)(468,498)(469,503)(470,504)(471,501)
(472,502)(473,507)(474,508)(475,505)(476,506)(477,511)(478,512)(479,509)
(480,510);
s1 := Sym(512)!(  1,257)(  2,258)(  3,259)(  4,260)(  5,262)(  6,261)(  7,264)
(  8,263)(  9,265)( 10,266)( 11,267)( 12,268)( 13,270)( 14,269)( 15,272)
( 16,271)( 17,275)( 18,276)( 19,273)( 20,274)( 21,280)( 22,279)( 23,278)
( 24,277)( 25,283)( 26,284)( 27,281)( 28,282)( 29,288)( 30,287)( 31,286)
( 32,285)( 33,305)( 34,306)( 35,307)( 36,308)( 37,310)( 38,309)( 39,312)
( 40,311)( 41,313)( 42,314)( 43,315)( 44,316)( 45,318)( 46,317)( 47,320)
( 48,319)( 49,289)( 50,290)( 51,291)( 52,292)( 53,294)( 54,293)( 55,296)
( 56,295)( 57,297)( 58,298)( 59,299)( 60,300)( 61,302)( 62,301)( 63,304)
( 64,303)( 65,329)( 66,330)( 67,331)( 68,332)( 69,334)( 70,333)( 71,336)
( 72,335)( 73,321)( 74,322)( 75,323)( 76,324)( 77,326)( 78,325)( 79,328)
( 80,327)( 81,347)( 82,348)( 83,345)( 84,346)( 85,352)( 86,351)( 87,350)
( 88,349)( 89,339)( 90,340)( 91,337)( 92,338)( 93,344)( 94,343)( 95,342)
( 96,341)( 97,377)( 98,378)( 99,379)(100,380)(101,382)(102,381)(103,384)
(104,383)(105,369)(106,370)(107,371)(108,372)(109,374)(110,373)(111,376)
(112,375)(113,361)(114,362)(115,363)(116,364)(117,366)(118,365)(119,368)
(120,367)(121,353)(122,354)(123,355)(124,356)(125,358)(126,357)(127,360)
(128,359)(129,401)(130,402)(131,403)(132,404)(133,406)(134,405)(135,408)
(136,407)(137,409)(138,410)(139,411)(140,412)(141,414)(142,413)(143,416)
(144,415)(145,385)(146,386)(147,387)(148,388)(149,390)(150,389)(151,392)
(152,391)(153,393)(154,394)(155,395)(156,396)(157,398)(158,397)(159,400)
(160,399)(161,417)(162,418)(163,419)(164,420)(165,422)(166,421)(167,424)
(168,423)(169,425)(170,426)(171,427)(172,428)(173,430)(174,429)(175,432)
(176,431)(177,435)(178,436)(179,433)(180,434)(181,440)(182,439)(183,438)
(184,437)(185,443)(186,444)(187,441)(188,442)(189,448)(190,447)(191,446)
(192,445)(193,473)(194,474)(195,475)(196,476)(197,478)(198,477)(199,480)
(200,479)(201,465)(202,466)(203,467)(204,468)(205,470)(206,469)(207,472)
(208,471)(209,457)(210,458)(211,459)(212,460)(213,462)(214,461)(215,464)
(216,463)(217,449)(218,450)(219,451)(220,452)(221,454)(222,453)(223,456)
(224,455)(225,489)(226,490)(227,491)(228,492)(229,494)(230,493)(231,496)
(232,495)(233,481)(234,482)(235,483)(236,484)(237,486)(238,485)(239,488)
(240,487)(241,507)(242,508)(243,505)(244,506)(245,512)(246,511)(247,510)
(248,509)(249,499)(250,500)(251,497)(252,498)(253,504)(254,503)(255,502)
(256,501);
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)!(  1,161)(  2,162)(  3,163)(  4,164)(  5,165)(  6,166)(  7,167)
(  8,168)(  9,170)( 10,169)( 11,172)( 12,171)( 13,174)( 14,173)( 15,176)
( 16,175)( 17,177)( 18,178)( 19,179)( 20,180)( 21,181)( 22,182)( 23,183)
( 24,184)( 25,186)( 26,185)( 27,188)( 28,187)( 29,190)( 30,189)( 31,192)
( 32,191)( 33,129)( 34,130)( 35,131)( 36,132)( 37,133)( 38,134)( 39,135)
( 40,136)( 41,138)( 42,137)( 43,140)( 44,139)( 45,142)( 46,141)( 47,144)
( 48,143)( 49,145)( 50,146)( 51,147)( 52,148)( 53,149)( 54,150)( 55,151)
( 56,152)( 57,154)( 58,153)( 59,156)( 60,155)( 61,158)( 62,157)( 63,160)
( 64,159)( 65,229)( 66,230)( 67,231)( 68,232)( 69,225)( 70,226)( 71,227)
( 72,228)( 73,238)( 74,237)( 75,240)( 76,239)( 77,234)( 78,233)( 79,236)
( 80,235)( 81,245)( 82,246)( 83,247)( 84,248)( 85,241)( 86,242)( 87,243)
( 88,244)( 89,254)( 90,253)( 91,256)( 92,255)( 93,250)( 94,249)( 95,252)
( 96,251)( 97,197)( 98,198)( 99,199)(100,200)(101,193)(102,194)(103,195)
(104,196)(105,206)(106,205)(107,208)(108,207)(109,202)(110,201)(111,204)
(112,203)(113,213)(114,214)(115,215)(116,216)(117,209)(118,210)(119,211)
(120,212)(121,222)(122,221)(123,224)(124,223)(125,218)(126,217)(127,220)
(128,219)(257,417)(258,418)(259,419)(260,420)(261,421)(262,422)(263,423)
(264,424)(265,426)(266,425)(267,428)(268,427)(269,430)(270,429)(271,432)
(272,431)(273,433)(274,434)(275,435)(276,436)(277,437)(278,438)(279,439)
(280,440)(281,442)(282,441)(283,444)(284,443)(285,446)(286,445)(287,448)
(288,447)(289,385)(290,386)(291,387)(292,388)(293,389)(294,390)(295,391)
(296,392)(297,394)(298,393)(299,396)(300,395)(301,398)(302,397)(303,400)
(304,399)(305,401)(306,402)(307,403)(308,404)(309,405)(310,406)(311,407)
(312,408)(313,410)(314,409)(315,412)(316,411)(317,414)(318,413)(319,416)
(320,415)(321,485)(322,486)(323,487)(324,488)(325,481)(326,482)(327,483)
(328,484)(329,494)(330,493)(331,496)(332,495)(333,490)(334,489)(335,492)
(336,491)(337,501)(338,502)(339,503)(340,504)(341,497)(342,498)(343,499)
(344,500)(345,510)(346,509)(347,512)(348,511)(349,506)(350,505)(351,508)
(352,507)(353,453)(354,454)(355,455)(356,456)(357,449)(358,450)(359,451)
(360,452)(361,462)(362,461)(363,464)(364,463)(365,458)(366,457)(367,460)
(368,459)(369,469)(370,470)(371,471)(372,472)(373,465)(374,466)(375,467)
(376,468)(377,478)(378,477)(379,480)(380,479)(381,474)(382,473)(383,476)
(384,475);
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*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

```
References : None.
to this polytope