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# Polytope of Type {8,8}

Atlas Canonical Name : {8,8}*512p
if this polytope has a name.
Group : SmallGroup(512,58354)
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
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {8,8}*256c, {8,4}*256b, {8,8}*256h
4-fold quotients : {4,8}*128a, {8,4}*128b
8-fold quotients : {4,8}*64a, {4,8}*64b, {4,4}*64
16-fold quotients : {4,4}*32, {2,8}*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, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 70)(  6, 69)(  7, 72)(  8, 71)
(  9, 73)( 10, 74)( 11, 75)( 12, 76)( 13, 78)( 14, 77)( 15, 80)( 16, 79)
( 17, 83)( 18, 84)( 19, 81)( 20, 82)( 21, 88)( 22, 87)( 23, 86)( 24, 85)
( 25, 91)( 26, 92)( 27, 89)( 28, 90)( 29, 96)( 30, 95)( 31, 94)( 32, 93)
( 33,101)( 34,102)( 35,103)( 36,104)( 37, 97)( 38, 98)( 39, 99)( 40,100)
( 41,109)( 42,110)( 43,111)( 44,112)( 45,105)( 46,106)( 47,107)( 48,108)
( 49,119)( 50,120)( 51,117)( 52,118)( 53,115)( 54,116)( 55,113)( 56,114)
( 57,127)( 58,128)( 59,125)( 60,126)( 61,123)( 62,124)( 63,121)( 64,122)
(129,193)(130,194)(131,195)(132,196)(133,198)(134,197)(135,200)(136,199)
(137,201)(138,202)(139,203)(140,204)(141,206)(142,205)(143,208)(144,207)
(145,211)(146,212)(147,209)(148,210)(149,216)(150,215)(151,214)(152,213)
(153,219)(154,220)(155,217)(156,218)(157,224)(158,223)(159,222)(160,221)
(161,229)(162,230)(163,231)(164,232)(165,225)(166,226)(167,227)(168,228)
(169,237)(170,238)(171,239)(172,240)(173,233)(174,234)(175,235)(176,236)
(177,247)(178,248)(179,245)(180,246)(181,243)(182,244)(183,241)(184,242)
(185,255)(186,256)(187,253)(188,254)(189,251)(190,252)(191,249)(192,250)
(257,321)(258,322)(259,323)(260,324)(261,326)(262,325)(263,328)(264,327)
(265,329)(266,330)(267,331)(268,332)(269,334)(270,333)(271,336)(272,335)
(273,339)(274,340)(275,337)(276,338)(277,344)(278,343)(279,342)(280,341)
(281,347)(282,348)(283,345)(284,346)(285,352)(286,351)(287,350)(288,349)
(289,357)(290,358)(291,359)(292,360)(293,353)(294,354)(295,355)(296,356)
(297,365)(298,366)(299,367)(300,368)(301,361)(302,362)(303,363)(304,364)
(305,375)(306,376)(307,373)(308,374)(309,371)(310,372)(311,369)(312,370)
(313,383)(314,384)(315,381)(316,382)(317,379)(318,380)(319,377)(320,378)
(385,449)(386,450)(387,451)(388,452)(389,454)(390,453)(391,456)(392,455)
(393,457)(394,458)(395,459)(396,460)(397,462)(398,461)(399,464)(400,463)
(401,467)(402,468)(403,465)(404,466)(405,472)(406,471)(407,470)(408,469)
(409,475)(410,476)(411,473)(412,474)(413,480)(414,479)(415,478)(416,477)
(417,485)(418,486)(419,487)(420,488)(421,481)(422,482)(423,483)(424,484)
(425,493)(426,494)(427,495)(428,496)(429,489)(430,490)(431,491)(432,492)
(433,503)(434,504)(435,501)(436,502)(437,499)(438,500)(439,497)(440,498)
(441,511)(442,512)(443,509)(444,510)(445,507)(446,508)(447,505)(448,506);;
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)
( 17,275)( 18,276)( 19,273)( 20,274)( 21,277)( 22,278)( 23,279)( 24,280)
( 25,283)( 26,284)( 27,281)( 28,282)( 29,285)( 30,286)( 31,287)( 32,288)
( 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)
( 49,315)( 50,316)( 51,313)( 52,314)( 53,317)( 54,318)( 55,319)( 56,320)
( 57,307)( 58,308)( 59,305)( 60,306)( 61,309)( 62,310)( 63,311)( 64,312)
( 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)
( 97,379)( 98,380)( 99,377)(100,378)(101,381)(102,382)(103,383)(104,384)
(105,371)(106,372)(107,369)(108,370)(109,373)(110,374)(111,375)(112,376)
(113,363)(114,364)(115,361)(116,362)(117,365)(118,366)(119,367)(120,368)
(121,355)(122,356)(123,353)(124,354)(125,357)(126,358)(127,359)(128,360)
(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)
(145,435)(146,436)(147,433)(148,434)(149,437)(150,438)(151,439)(152,440)
(153,443)(154,444)(155,441)(156,442)(157,445)(158,446)(159,447)(160,448)
(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)
(177,403)(178,404)(179,401)(180,402)(181,405)(182,406)(183,407)(184,408)
(185,411)(186,412)(187,409)(188,410)(189,413)(190,414)(191,415)(192,416)
(193,501)(194,502)(195,503)(196,504)(197,500)(198,499)(199,498)(200,497)
(201,509)(202,510)(203,511)(204,512)(205,508)(206,507)(207,506)(208,505)
(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)
(241,456)(242,455)(243,454)(244,453)(245,449)(246,450)(247,451)(248,452)
(249,464)(250,463)(251,462)(252,461)(253,457)(254,458)(255,459)(256,460);;
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, s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope