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# Polytope of Type {8,8}

Atlas Canonical Name : {8,8}*512o
if this polytope has a name.
Group : SmallGroup(512,58342)
Rank : 3
Schlafli Type : {8,8}
Number of vertices, edges, etc : 32, 128, 32
Order of s0s1s2 : 8
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {8,8}*256b, {8,4}*256d, {8,8}*256h
4-fold quotients : {4,8}*128b, {8,4}*128b
8-fold quotients : {4,4}*64
16-fold quotients : {4,4}*32
32-fold quotients : {2,4}*16, {4,2}*16
64-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,193)(  2,194)(  3,195)(  4,196)(  5,198)(  6,197)(  7,200)(  8,199)
(  9,201)( 10,202)( 11,203)( 12,204)( 13,206)( 14,205)( 15,208)( 16,207)
( 17,215)( 18,216)( 19,213)( 20,214)( 21,211)( 22,212)( 23,209)( 24,210)
( 25,223)( 26,224)( 27,221)( 28,222)( 29,219)( 30,220)( 31,217)( 32,218)
( 33,237)( 34,238)( 35,239)( 36,240)( 37,233)( 38,234)( 39,235)( 40,236)
( 41,229)( 42,230)( 43,231)( 44,232)( 45,225)( 46,226)( 47,227)( 48,228)
( 49,251)( 50,252)( 51,249)( 52,250)( 53,256)( 54,255)( 55,254)( 56,253)
( 57,243)( 58,244)( 59,241)( 60,242)( 61,248)( 62,247)( 63,246)( 64,245)
( 65,129)( 66,130)( 67,131)( 68,132)( 69,134)( 70,133)( 71,136)( 72,135)
( 73,137)( 74,138)( 75,139)( 76,140)( 77,142)( 78,141)( 79,144)( 80,143)
( 81,151)( 82,152)( 83,149)( 84,150)( 85,147)( 86,148)( 87,145)( 88,146)
( 89,159)( 90,160)( 91,157)( 92,158)( 93,155)( 94,156)( 95,153)( 96,154)
( 97,173)( 98,174)( 99,175)(100,176)(101,169)(102,170)(103,171)(104,172)
(105,165)(106,166)(107,167)(108,168)(109,161)(110,162)(111,163)(112,164)
(113,187)(114,188)(115,185)(116,186)(117,192)(118,191)(119,190)(120,189)
(121,179)(122,180)(123,177)(124,178)(125,184)(126,183)(127,182)(128,181)
(257,449)(258,450)(259,451)(260,452)(261,454)(262,453)(263,456)(264,455)
(265,457)(266,458)(267,459)(268,460)(269,462)(270,461)(271,464)(272,463)
(273,471)(274,472)(275,469)(276,470)(277,467)(278,468)(279,465)(280,466)
(281,479)(282,480)(283,477)(284,478)(285,475)(286,476)(287,473)(288,474)
(289,493)(290,494)(291,495)(292,496)(293,489)(294,490)(295,491)(296,492)
(297,485)(298,486)(299,487)(300,488)(301,481)(302,482)(303,483)(304,484)
(305,507)(306,508)(307,505)(308,506)(309,512)(310,511)(311,510)(312,509)
(313,499)(314,500)(315,497)(316,498)(317,504)(318,503)(319,502)(320,501)
(321,385)(322,386)(323,387)(324,388)(325,390)(326,389)(327,392)(328,391)
(329,393)(330,394)(331,395)(332,396)(333,398)(334,397)(335,400)(336,399)
(337,407)(338,408)(339,405)(340,406)(341,403)(342,404)(343,401)(344,402)
(345,415)(346,416)(347,413)(348,414)(349,411)(350,412)(351,409)(352,410)
(353,429)(354,430)(355,431)(356,432)(357,425)(358,426)(359,427)(360,428)
(361,421)(362,422)(363,423)(364,424)(365,417)(366,418)(367,419)(368,420)
(369,443)(370,444)(371,441)(372,442)(373,448)(374,447)(375,446)(376,445)
(377,435)(378,436)(379,433)(380,434)(381,440)(382,439)(383,438)(384,437);;
s1 := (  1,257)(  2,258)(  3,259)(  4,260)(  5,263)(  6,264)(  7,261)(  8,262)
(  9,265)( 10,266)( 11,267)( 12,268)( 13,271)( 14,272)( 15,269)( 16,270)
( 17,274)( 18,273)( 19,276)( 20,275)( 21,280)( 22,279)( 23,278)( 24,277)
( 25,282)( 26,281)( 27,284)( 28,283)( 29,288)( 30,287)( 31,286)( 32,285)
( 33,297)( 34,298)( 35,299)( 36,300)( 37,303)( 38,304)( 39,301)( 40,302)
( 41,289)( 42,290)( 43,291)( 44,292)( 45,295)( 46,296)( 47,293)( 48,294)
( 49,314)( 50,313)( 51,316)( 52,315)( 53,320)( 54,319)( 55,318)( 56,317)
( 57,306)( 58,305)( 59,308)( 60,307)( 61,312)( 62,311)( 63,310)( 64,309)
( 65,337)( 66,338)( 67,339)( 68,340)( 69,343)( 70,344)( 71,341)( 72,342)
( 73,345)( 74,346)( 75,347)( 76,348)( 77,351)( 78,352)( 79,349)( 80,350)
( 81,321)( 82,322)( 83,323)( 84,324)( 85,327)( 86,328)( 87,325)( 88,326)
( 89,329)( 90,330)( 91,331)( 92,332)( 93,335)( 94,336)( 95,333)( 96,334)
( 97,380)( 98,379)( 99,378)(100,377)(101,382)(102,381)(103,384)(104,383)
(105,372)(106,371)(107,370)(108,369)(109,374)(110,373)(111,376)(112,375)
(113,364)(114,363)(115,362)(116,361)(117,366)(118,365)(119,368)(120,367)
(121,356)(122,355)(123,354)(124,353)(125,358)(126,357)(127,360)(128,359)
(129,417)(130,418)(131,419)(132,420)(133,423)(134,424)(135,421)(136,422)
(137,425)(138,426)(139,427)(140,428)(141,431)(142,432)(143,429)(144,430)
(145,434)(146,433)(147,436)(148,435)(149,440)(150,439)(151,438)(152,437)
(153,442)(154,441)(155,444)(156,443)(157,448)(158,447)(159,446)(160,445)
(161,385)(162,386)(163,387)(164,388)(165,391)(166,392)(167,389)(168,390)
(169,393)(170,394)(171,395)(172,396)(173,399)(174,400)(175,397)(176,398)
(177,402)(178,401)(179,404)(180,403)(181,408)(182,407)(183,406)(184,405)
(185,410)(186,409)(187,412)(188,411)(189,416)(190,415)(191,414)(192,413)
(193,502)(194,501)(195,504)(196,503)(197,499)(198,500)(199,497)(200,498)
(201,510)(202,509)(203,512)(204,511)(205,507)(206,508)(207,505)(208,506)
(209,485)(210,486)(211,487)(212,488)(213,484)(214,483)(215,482)(216,481)
(217,493)(218,494)(219,495)(220,496)(221,492)(222,491)(223,490)(224,489)
(225,472)(226,471)(227,470)(228,469)(229,465)(230,466)(231,467)(232,468)
(233,480)(234,479)(235,478)(236,477)(237,473)(238,474)(239,475)(240,476)
(241,455)(242,456)(243,453)(244,454)(245,450)(246,449)(247,452)(248,451)
(249,463)(250,464)(251,461)(252,462)(253,458)(254,457)(255,460)(256,459);;
s2 := (  1,129)(  2,130)(  3,132)(  4,131)(  5,133)(  6,134)(  7,136)(  8,135)
(  9,137)( 10,138)( 11,140)( 12,139)( 13,141)( 14,142)( 15,144)( 16,143)
( 17,152)( 18,151)( 19,149)( 20,150)( 21,147)( 22,148)( 23,146)( 24,145)
( 25,160)( 26,159)( 27,157)( 28,158)( 29,155)( 30,156)( 31,154)( 32,153)
( 33,169)( 34,170)( 35,172)( 36,171)( 37,173)( 38,174)( 39,176)( 40,175)
( 41,161)( 42,162)( 43,164)( 44,163)( 45,165)( 46,166)( 47,168)( 48,167)
( 49,192)( 50,191)( 51,189)( 52,190)( 53,187)( 54,188)( 55,186)( 56,185)
( 57,184)( 58,183)( 59,181)( 60,182)( 61,179)( 62,180)( 63,178)( 64,177)
( 65,193)( 66,194)( 67,196)( 68,195)( 69,197)( 70,198)( 71,200)( 72,199)
( 73,201)( 74,202)( 75,204)( 76,203)( 77,205)( 78,206)( 79,208)( 80,207)
( 81,216)( 82,215)( 83,213)( 84,214)( 85,211)( 86,212)( 87,210)( 88,209)
( 89,224)( 90,223)( 91,221)( 92,222)( 93,219)( 94,220)( 95,218)( 96,217)
( 97,233)( 98,234)( 99,236)(100,235)(101,237)(102,238)(103,240)(104,239)
(105,225)(106,226)(107,228)(108,227)(109,229)(110,230)(111,232)(112,231)
(113,256)(114,255)(115,253)(116,254)(117,251)(118,252)(119,250)(120,249)
(121,248)(122,247)(123,245)(124,246)(125,243)(126,244)(127,242)(128,241)
(257,385)(258,386)(259,388)(260,387)(261,389)(262,390)(263,392)(264,391)
(265,393)(266,394)(267,396)(268,395)(269,397)(270,398)(271,400)(272,399)
(273,408)(274,407)(275,405)(276,406)(277,403)(278,404)(279,402)(280,401)
(281,416)(282,415)(283,413)(284,414)(285,411)(286,412)(287,410)(288,409)
(289,425)(290,426)(291,428)(292,427)(293,429)(294,430)(295,432)(296,431)
(297,417)(298,418)(299,420)(300,419)(301,421)(302,422)(303,424)(304,423)
(305,448)(306,447)(307,445)(308,446)(309,443)(310,444)(311,442)(312,441)
(313,440)(314,439)(315,437)(316,438)(317,435)(318,436)(319,434)(320,433)
(321,449)(322,450)(323,452)(324,451)(325,453)(326,454)(327,456)(328,455)
(329,457)(330,458)(331,460)(332,459)(333,461)(334,462)(335,464)(336,463)
(337,472)(338,471)(339,469)(340,470)(341,467)(342,468)(343,466)(344,465)
(345,480)(346,479)(347,477)(348,478)(349,475)(350,476)(351,474)(352,473)
(353,489)(354,490)(355,492)(356,491)(357,493)(358,494)(359,496)(360,495)
(361,481)(362,482)(363,484)(364,483)(365,485)(366,486)(367,488)(368,487)
(369,512)(370,511)(371,509)(372,510)(373,507)(374,508)(375,506)(376,505)
(377,504)(378,503)(379,501)(380,502)(381,499)(382,500)(383,498)(384,497);;
poly := Group([s0,s1,s2]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1 >;

```
References : None.
to this polytope