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# Polytope of Type {8,32}

Atlas Canonical Name : {8,32}*512a
if this polytope has a name.
Group : SmallGroup(512,58708)
Rank : 3
Schlafli Type : {8,32}
Number of vertices, edges, etc : 8, 128, 32
Order of s0s1s2 : 32
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
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 : {8,16}*256c, {4,32}*256a
4-fold quotients : {8,8}*128a, {4,16}*128a, {2,32}*128
8-fold quotients : {4,8}*64a, {8,4}*64b, {2,16}*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, 69)(  6, 70)(  7, 71)(  8, 72)
(  9, 73)( 10, 74)( 11, 75)( 12, 76)( 13, 77)( 14, 78)( 15, 79)( 16, 80)
( 17, 85)( 18, 86)( 19, 87)( 20, 88)( 21, 81)( 22, 82)( 23, 83)( 24, 84)
( 25, 93)( 26, 94)( 27, 95)( 28, 96)( 29, 89)( 30, 90)( 31, 91)( 32, 92)
( 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)
( 49,113)( 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)
(145,213)(146,214)(147,215)(148,216)(149,209)(150,210)(151,211)(152,212)
(153,221)(154,222)(155,223)(156,224)(157,217)(158,218)(159,219)(160,220)
(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)
(177,241)(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)
(273,341)(274,342)(275,343)(276,344)(277,337)(278,338)(279,339)(280,340)
(281,349)(282,350)(283,351)(284,352)(285,345)(286,346)(287,347)(288,348)
(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)
(305,369)(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)
(401,469)(402,470)(403,471)(404,472)(405,465)(406,466)(407,467)(408,468)
(409,477)(410,478)(411,479)(412,480)(413,473)(414,474)(415,475)(416,476)
(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)
(433,497)(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);;
s1 := (  1,257)(  2,258)(  3,260)(  4,259)(  5,261)(  6,262)(  7,264)(  8,263)
(  9,267)( 10,268)( 11,265)( 12,266)( 13,271)( 14,272)( 15,269)( 16,270)
( 17,277)( 18,278)( 19,280)( 20,279)( 21,273)( 22,274)( 23,276)( 24,275)
( 25,287)( 26,288)( 27,285)( 28,286)( 29,283)( 30,284)( 31,281)( 32,282)
( 33,297)( 34,298)( 35,300)( 36,299)( 37,301)( 38,302)( 39,304)( 40,303)
( 41,289)( 42,290)( 43,292)( 44,291)( 45,293)( 46,294)( 47,296)( 48,295)
( 49,317)( 50,318)( 51,320)( 52,319)( 53,313)( 54,314)( 55,316)( 56,315)
( 57,309)( 58,310)( 59,312)( 60,311)( 61,305)( 62,306)( 63,308)( 64,307)
( 65,337)( 66,338)( 67,340)( 68,339)( 69,341)( 70,342)( 71,344)( 72,343)
( 73,347)( 74,348)( 75,345)( 76,346)( 77,351)( 78,352)( 79,349)( 80,350)
( 81,321)( 82,322)( 83,324)( 84,323)( 85,325)( 86,326)( 87,328)( 88,327)
( 89,331)( 90,332)( 91,329)( 92,330)( 93,335)( 94,336)( 95,333)( 96,334)
( 97,377)( 98,378)( 99,380)(100,379)(101,381)(102,382)(103,384)(104,383)
(105,369)(106,370)(107,372)(108,371)(109,373)(110,374)(111,376)(112,375)
(113,361)(114,362)(115,364)(116,363)(117,365)(118,366)(119,368)(120,367)
(121,353)(122,354)(123,356)(124,355)(125,357)(126,358)(127,360)(128,359)
(129,417)(130,418)(131,420)(132,419)(133,421)(134,422)(135,424)(136,423)
(137,427)(138,428)(139,425)(140,426)(141,431)(142,432)(143,429)(144,430)
(145,437)(146,438)(147,440)(148,439)(149,433)(150,434)(151,436)(152,435)
(153,447)(154,448)(155,445)(156,446)(157,443)(158,444)(159,441)(160,442)
(161,385)(162,386)(163,388)(164,387)(165,389)(166,390)(167,392)(168,391)
(169,395)(170,396)(171,393)(172,394)(173,399)(174,400)(175,397)(176,398)
(177,405)(178,406)(179,408)(180,407)(181,401)(182,402)(183,404)(184,403)
(185,415)(186,416)(187,413)(188,414)(189,411)(190,412)(191,409)(192,410)
(193,501)(194,502)(195,504)(196,503)(197,497)(198,498)(199,500)(200,499)
(201,511)(202,512)(203,509)(204,510)(205,507)(206,508)(207,505)(208,506)
(209,485)(210,486)(211,488)(212,487)(213,481)(214,482)(215,484)(216,483)
(217,495)(218,496)(219,493)(220,494)(221,491)(222,492)(223,489)(224,490)
(225,469)(226,470)(227,472)(228,471)(229,465)(230,466)(231,468)(232,467)
(233,479)(234,480)(235,477)(236,478)(237,475)(238,476)(239,473)(240,474)
(241,453)(242,454)(243,456)(244,455)(245,449)(246,450)(247,452)(248,451)
(249,463)(250,464)(251,461)(252,462)(253,459)(254,460)(255,457)(256,458);;
s2 := (  1,129)(  2,130)(  3,132)(  4,131)(  5,133)(  6,134)(  7,136)(  8,135)
(  9,139)( 10,140)( 11,137)( 12,138)( 13,143)( 14,144)( 15,141)( 16,142)
( 17,149)( 18,150)( 19,152)( 20,151)( 21,145)( 22,146)( 23,148)( 24,147)
( 25,159)( 26,160)( 27,157)( 28,158)( 29,155)( 30,156)( 31,153)( 32,154)
( 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)
( 49,189)( 50,190)( 51,192)( 52,191)( 53,185)( 54,186)( 55,188)( 56,187)
( 57,181)( 58,182)( 59,184)( 60,183)( 61,177)( 62,178)( 63,180)( 64,179)
( 65,193)( 66,194)( 67,196)( 68,195)( 69,197)( 70,198)( 71,200)( 72,199)
( 73,203)( 74,204)( 75,201)( 76,202)( 77,207)( 78,208)( 79,205)( 80,206)
( 81,213)( 82,214)( 83,216)( 84,215)( 85,209)( 86,210)( 87,212)( 88,211)
( 89,223)( 90,224)( 91,221)( 92,222)( 93,219)( 94,220)( 95,217)( 96,218)
( 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)
(113,253)(114,254)(115,256)(116,255)(117,249)(118,250)(119,252)(120,251)
(121,245)(122,246)(123,248)(124,247)(125,241)(126,242)(127,244)(128,243)
(257,385)(258,386)(259,388)(260,387)(261,389)(262,390)(263,392)(264,391)
(265,395)(266,396)(267,393)(268,394)(269,399)(270,400)(271,397)(272,398)
(273,405)(274,406)(275,408)(276,407)(277,401)(278,402)(279,404)(280,403)
(281,415)(282,416)(283,413)(284,414)(285,411)(286,412)(287,409)(288,410)
(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)
(305,445)(306,446)(307,448)(308,447)(309,441)(310,442)(311,444)(312,443)
(313,437)(314,438)(315,440)(316,439)(317,433)(318,434)(319,436)(320,435)
(321,449)(322,450)(323,452)(324,451)(325,453)(326,454)(327,456)(328,455)
(329,459)(330,460)(331,457)(332,458)(333,463)(334,464)(335,461)(336,462)
(337,469)(338,470)(339,472)(340,471)(341,465)(342,466)(343,468)(344,467)
(345,479)(346,480)(347,477)(348,478)(349,475)(350,476)(351,473)(352,474)
(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)
(369,509)(370,510)(371,512)(372,511)(373,505)(374,506)(375,508)(376,507)
(377,501)(378,502)(379,504)(380,503)(381,497)(382,498)(383,500)(384,499);;
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*s0*s1*s2*s0*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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, 69)(  6, 70)(  7, 71)
(  8, 72)(  9, 73)( 10, 74)( 11, 75)( 12, 76)( 13, 77)( 14, 78)( 15, 79)
( 16, 80)( 17, 85)( 18, 86)( 19, 87)( 20, 88)( 21, 81)( 22, 82)( 23, 83)
( 24, 84)( 25, 93)( 26, 94)( 27, 95)( 28, 96)( 29, 89)( 30, 90)( 31, 91)
( 32, 92)( 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)( 49,113)( 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)(145,213)(146,214)(147,215)(148,216)(149,209)(150,210)(151,211)
(152,212)(153,221)(154,222)(155,223)(156,224)(157,217)(158,218)(159,219)
(160,220)(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)(177,241)(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)(273,341)(274,342)(275,343)(276,344)(277,337)(278,338)(279,339)
(280,340)(281,349)(282,350)(283,351)(284,352)(285,345)(286,346)(287,347)
(288,348)(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)(305,369)(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)(401,469)(402,470)(403,471)(404,472)(405,465)(406,466)(407,467)
(408,468)(409,477)(410,478)(411,479)(412,480)(413,473)(414,474)(415,475)
(416,476)(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)(433,497)(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);
s1 := Sym(512)!(  1,257)(  2,258)(  3,260)(  4,259)(  5,261)(  6,262)(  7,264)
(  8,263)(  9,267)( 10,268)( 11,265)( 12,266)( 13,271)( 14,272)( 15,269)
( 16,270)( 17,277)( 18,278)( 19,280)( 20,279)( 21,273)( 22,274)( 23,276)
( 24,275)( 25,287)( 26,288)( 27,285)( 28,286)( 29,283)( 30,284)( 31,281)
( 32,282)( 33,297)( 34,298)( 35,300)( 36,299)( 37,301)( 38,302)( 39,304)
( 40,303)( 41,289)( 42,290)( 43,292)( 44,291)( 45,293)( 46,294)( 47,296)
( 48,295)( 49,317)( 50,318)( 51,320)( 52,319)( 53,313)( 54,314)( 55,316)
( 56,315)( 57,309)( 58,310)( 59,312)( 60,311)( 61,305)( 62,306)( 63,308)
( 64,307)( 65,337)( 66,338)( 67,340)( 68,339)( 69,341)( 70,342)( 71,344)
( 72,343)( 73,347)( 74,348)( 75,345)( 76,346)( 77,351)( 78,352)( 79,349)
( 80,350)( 81,321)( 82,322)( 83,324)( 84,323)( 85,325)( 86,326)( 87,328)
( 88,327)( 89,331)( 90,332)( 91,329)( 92,330)( 93,335)( 94,336)( 95,333)
( 96,334)( 97,377)( 98,378)( 99,380)(100,379)(101,381)(102,382)(103,384)
(104,383)(105,369)(106,370)(107,372)(108,371)(109,373)(110,374)(111,376)
(112,375)(113,361)(114,362)(115,364)(116,363)(117,365)(118,366)(119,368)
(120,367)(121,353)(122,354)(123,356)(124,355)(125,357)(126,358)(127,360)
(128,359)(129,417)(130,418)(131,420)(132,419)(133,421)(134,422)(135,424)
(136,423)(137,427)(138,428)(139,425)(140,426)(141,431)(142,432)(143,429)
(144,430)(145,437)(146,438)(147,440)(148,439)(149,433)(150,434)(151,436)
(152,435)(153,447)(154,448)(155,445)(156,446)(157,443)(158,444)(159,441)
(160,442)(161,385)(162,386)(163,388)(164,387)(165,389)(166,390)(167,392)
(168,391)(169,395)(170,396)(171,393)(172,394)(173,399)(174,400)(175,397)
(176,398)(177,405)(178,406)(179,408)(180,407)(181,401)(182,402)(183,404)
(184,403)(185,415)(186,416)(187,413)(188,414)(189,411)(190,412)(191,409)
(192,410)(193,501)(194,502)(195,504)(196,503)(197,497)(198,498)(199,500)
(200,499)(201,511)(202,512)(203,509)(204,510)(205,507)(206,508)(207,505)
(208,506)(209,485)(210,486)(211,488)(212,487)(213,481)(214,482)(215,484)
(216,483)(217,495)(218,496)(219,493)(220,494)(221,491)(222,492)(223,489)
(224,490)(225,469)(226,470)(227,472)(228,471)(229,465)(230,466)(231,468)
(232,467)(233,479)(234,480)(235,477)(236,478)(237,475)(238,476)(239,473)
(240,474)(241,453)(242,454)(243,456)(244,455)(245,449)(246,450)(247,452)
(248,451)(249,463)(250,464)(251,461)(252,462)(253,459)(254,460)(255,457)
(256,458);
s2 := Sym(512)!(  1,129)(  2,130)(  3,132)(  4,131)(  5,133)(  6,134)(  7,136)
(  8,135)(  9,139)( 10,140)( 11,137)( 12,138)( 13,143)( 14,144)( 15,141)
( 16,142)( 17,149)( 18,150)( 19,152)( 20,151)( 21,145)( 22,146)( 23,148)
( 24,147)( 25,159)( 26,160)( 27,157)( 28,158)( 29,155)( 30,156)( 31,153)
( 32,154)( 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)( 49,189)( 50,190)( 51,192)( 52,191)( 53,185)( 54,186)( 55,188)
( 56,187)( 57,181)( 58,182)( 59,184)( 60,183)( 61,177)( 62,178)( 63,180)
( 64,179)( 65,193)( 66,194)( 67,196)( 68,195)( 69,197)( 70,198)( 71,200)
( 72,199)( 73,203)( 74,204)( 75,201)( 76,202)( 77,207)( 78,208)( 79,205)
( 80,206)( 81,213)( 82,214)( 83,216)( 84,215)( 85,209)( 86,210)( 87,212)
( 88,211)( 89,223)( 90,224)( 91,221)( 92,222)( 93,219)( 94,220)( 95,217)
( 96,218)( 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)(113,253)(114,254)(115,256)(116,255)(117,249)(118,250)(119,252)
(120,251)(121,245)(122,246)(123,248)(124,247)(125,241)(126,242)(127,244)
(128,243)(257,385)(258,386)(259,388)(260,387)(261,389)(262,390)(263,392)
(264,391)(265,395)(266,396)(267,393)(268,394)(269,399)(270,400)(271,397)
(272,398)(273,405)(274,406)(275,408)(276,407)(277,401)(278,402)(279,404)
(280,403)(281,415)(282,416)(283,413)(284,414)(285,411)(286,412)(287,409)
(288,410)(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)(305,445)(306,446)(307,448)(308,447)(309,441)(310,442)(311,444)
(312,443)(313,437)(314,438)(315,440)(316,439)(317,433)(318,434)(319,436)
(320,435)(321,449)(322,450)(323,452)(324,451)(325,453)(326,454)(327,456)
(328,455)(329,459)(330,460)(331,457)(332,458)(333,463)(334,464)(335,461)
(336,462)(337,469)(338,470)(339,472)(340,471)(341,465)(342,466)(343,468)
(344,467)(345,479)(346,480)(347,477)(348,478)(349,475)(350,476)(351,473)
(352,474)(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)(369,509)(370,510)(371,512)(372,511)(373,505)(374,506)(375,508)
(376,507)(377,501)(378,502)(379,504)(380,503)(381,497)(382,498)(383,500)
(384,499);
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*s0*s1*s2*s0*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope