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Polytope of Type {8,16}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,16}*512c
if this polytope has a name.
Group : SmallGroup(512,32832)
Rank : 3
Schlafli Type : {8,16}
Number of vertices, edges, etc : 16, 128, 32
Order of s0s1s2 : 16
Order of s0s1s2s1 : 4
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}*256a, {4,16}*256a, {8,16}*256c, {8,16}*256d, {8,16}*256e, {8,16}*256f
   4-fold quotients : {4,8}*128a, {8,4}*128a, {8,8}*128a, {8,8}*128b, {8,8}*128c, {8,8}*128d, {4,16}*128a, {4,16}*128b
   8-fold quotients : {4,8}*64a, {8,4}*64a, {4,8}*64b, {8,4}*64b, {4,4}*64, {2,16}*64
   16-fold quotients : {4,4}*32, {2,8}*32, {8,2}*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, 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)
( 33,101)( 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)(133,197)(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)
(161,229)(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)(261,325)(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)
(289,357)(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)(389,453)(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)
(417,485)(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)(  5,261)(  6,262)(  7,263)(  8,264)
(  9,266)( 10,265)( 11,268)( 12,267)( 13,270)( 14,269)( 15,272)( 16,271)
( 17,275)( 18,276)( 19,273)( 20,274)( 21,279)( 22,280)( 23,277)( 24,278)
( 25,284)( 26,283)( 27,282)( 28,281)( 29,288)( 30,287)( 31,286)( 32,285)
( 33,297)( 34,298)( 35,299)( 36,300)( 37,301)( 38,302)( 39,303)( 40,304)
( 41,289)( 42,290)( 43,291)( 44,292)( 45,293)( 46,294)( 47,295)( 48,296)
( 49,315)( 50,316)( 51,313)( 52,314)( 53,319)( 54,320)( 55,317)( 56,318)
( 57,307)( 58,308)( 59,305)( 60,306)( 61,311)( 62,312)( 63,309)( 64,310)
( 65,337)( 66,338)( 67,339)( 68,340)( 69,341)( 70,342)( 71,343)( 72,344)
( 73,346)( 74,345)( 75,348)( 76,347)( 77,350)( 78,349)( 79,352)( 80,351)
( 81,321)( 82,322)( 83,323)( 84,324)( 85,325)( 86,326)( 87,327)( 88,328)
( 89,330)( 90,329)( 91,332)( 92,331)( 93,334)( 94,333)( 95,336)( 96,335)
( 97,377)( 98,378)( 99,379)(100,380)(101,381)(102,382)(103,383)(104,384)
(105,369)(106,370)(107,371)(108,372)(109,373)(110,374)(111,375)(112,376)
(113,361)(114,362)(115,363)(116,364)(117,365)(118,366)(119,367)(120,368)
(121,353)(122,354)(123,355)(124,356)(125,357)(126,358)(127,359)(128,360)
(129,417)(130,418)(131,419)(132,420)(133,421)(134,422)(135,423)(136,424)
(137,426)(138,425)(139,428)(140,427)(141,430)(142,429)(143,432)(144,431)
(145,435)(146,436)(147,433)(148,434)(149,439)(150,440)(151,437)(152,438)
(153,444)(154,443)(155,442)(156,441)(157,448)(158,447)(159,446)(160,445)
(161,385)(162,386)(163,387)(164,388)(165,389)(166,390)(167,391)(168,392)
(169,394)(170,393)(171,396)(172,395)(173,398)(174,397)(175,400)(176,399)
(177,403)(178,404)(179,401)(180,402)(181,407)(182,408)(183,405)(184,406)
(185,412)(186,411)(187,410)(188,409)(189,416)(190,415)(191,414)(192,413)
(193,501)(194,502)(195,503)(196,504)(197,497)(198,498)(199,499)(200,500)
(201,510)(202,509)(203,512)(204,511)(205,506)(206,505)(207,508)(208,507)
(209,485)(210,486)(211,487)(212,488)(213,481)(214,482)(215,483)(216,484)
(217,494)(218,493)(219,496)(220,495)(221,490)(222,489)(223,492)(224,491)
(225,469)(226,470)(227,471)(228,472)(229,465)(230,466)(231,467)(232,468)
(233,478)(234,477)(235,480)(236,479)(237,474)(238,473)(239,476)(240,475)
(241,453)(242,454)(243,455)(244,456)(245,449)(246,450)(247,451)(248,452)
(249,462)(250,461)(251,464)(252,463)(253,458)(254,457)(255,460)(256,459);;
s2 := (  1,129)(  2,130)(  3,131)(  4,132)(  5,133)(  6,134)(  7,135)(  8,136)
(  9,138)( 10,137)( 11,140)( 12,139)( 13,142)( 14,141)( 15,144)( 16,143)
( 17,149)( 18,150)( 19,151)( 20,152)( 21,145)( 22,146)( 23,147)( 24,148)
( 25,158)( 26,157)( 27,160)( 28,159)( 29,154)( 30,153)( 31,156)( 32,155)
( 33,169)( 34,170)( 35,171)( 36,172)( 37,173)( 38,174)( 39,175)( 40,176)
( 41,161)( 42,162)( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)( 48,168)
( 49,189)( 50,190)( 51,191)( 52,192)( 53,185)( 54,186)( 55,187)( 56,188)
( 57,181)( 58,182)( 59,183)( 60,184)( 61,177)( 62,178)( 63,179)( 64,180)
( 65,193)( 66,194)( 67,195)( 68,196)( 69,197)( 70,198)( 71,199)( 72,200)
( 73,202)( 74,201)( 75,204)( 76,203)( 77,206)( 78,205)( 79,208)( 80,207)
( 81,213)( 82,214)( 83,215)( 84,216)( 85,209)( 86,210)( 87,211)( 88,212)
( 89,222)( 90,221)( 91,224)( 92,223)( 93,218)( 94,217)( 95,220)( 96,219)
( 97,233)( 98,234)( 99,235)(100,236)(101,237)(102,238)(103,239)(104,240)
(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)(112,232)
(113,253)(114,254)(115,255)(116,256)(117,249)(118,250)(119,251)(120,252)
(121,245)(122,246)(123,247)(124,248)(125,241)(126,242)(127,243)(128,244)
(257,385)(258,386)(259,387)(260,388)(261,389)(262,390)(263,391)(264,392)
(265,394)(266,393)(267,396)(268,395)(269,398)(270,397)(271,400)(272,399)
(273,405)(274,406)(275,407)(276,408)(277,401)(278,402)(279,403)(280,404)
(281,414)(282,413)(283,416)(284,415)(285,410)(286,409)(287,412)(288,411)
(289,425)(290,426)(291,427)(292,428)(293,429)(294,430)(295,431)(296,432)
(297,417)(298,418)(299,419)(300,420)(301,421)(302,422)(303,423)(304,424)
(305,445)(306,446)(307,447)(308,448)(309,441)(310,442)(311,443)(312,444)
(313,437)(314,438)(315,439)(316,440)(317,433)(318,434)(319,435)(320,436)
(321,449)(322,450)(323,451)(324,452)(325,453)(326,454)(327,455)(328,456)
(329,458)(330,457)(331,460)(332,459)(333,462)(334,461)(335,464)(336,463)
(337,469)(338,470)(339,471)(340,472)(341,465)(342,466)(343,467)(344,468)
(345,478)(346,477)(347,480)(348,479)(349,474)(350,473)(351,476)(352,475)
(353,489)(354,490)(355,491)(356,492)(357,493)(358,494)(359,495)(360,496)
(361,481)(362,482)(363,483)(364,484)(365,485)(366,486)(367,487)(368,488)
(369,509)(370,510)(371,511)(372,512)(373,505)(374,506)(375,507)(376,508)
(377,501)(378,502)(379,503)(380,504)(381,497)(382,498)(383,499)(384,500);;
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*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*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 ];;
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, 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)( 33,101)( 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)(133,197)(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)(161,229)(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)(261,325)(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)(289,357)(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)(389,453)(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)(417,485)(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)(  5,261)(  6,262)(  7,263)
(  8,264)(  9,266)( 10,265)( 11,268)( 12,267)( 13,270)( 14,269)( 15,272)
( 16,271)( 17,275)( 18,276)( 19,273)( 20,274)( 21,279)( 22,280)( 23,277)
( 24,278)( 25,284)( 26,283)( 27,282)( 28,281)( 29,288)( 30,287)( 31,286)
( 32,285)( 33,297)( 34,298)( 35,299)( 36,300)( 37,301)( 38,302)( 39,303)
( 40,304)( 41,289)( 42,290)( 43,291)( 44,292)( 45,293)( 46,294)( 47,295)
( 48,296)( 49,315)( 50,316)( 51,313)( 52,314)( 53,319)( 54,320)( 55,317)
( 56,318)( 57,307)( 58,308)( 59,305)( 60,306)( 61,311)( 62,312)( 63,309)
( 64,310)( 65,337)( 66,338)( 67,339)( 68,340)( 69,341)( 70,342)( 71,343)
( 72,344)( 73,346)( 74,345)( 75,348)( 76,347)( 77,350)( 78,349)( 79,352)
( 80,351)( 81,321)( 82,322)( 83,323)( 84,324)( 85,325)( 86,326)( 87,327)
( 88,328)( 89,330)( 90,329)( 91,332)( 92,331)( 93,334)( 94,333)( 95,336)
( 96,335)( 97,377)( 98,378)( 99,379)(100,380)(101,381)(102,382)(103,383)
(104,384)(105,369)(106,370)(107,371)(108,372)(109,373)(110,374)(111,375)
(112,376)(113,361)(114,362)(115,363)(116,364)(117,365)(118,366)(119,367)
(120,368)(121,353)(122,354)(123,355)(124,356)(125,357)(126,358)(127,359)
(128,360)(129,417)(130,418)(131,419)(132,420)(133,421)(134,422)(135,423)
(136,424)(137,426)(138,425)(139,428)(140,427)(141,430)(142,429)(143,432)
(144,431)(145,435)(146,436)(147,433)(148,434)(149,439)(150,440)(151,437)
(152,438)(153,444)(154,443)(155,442)(156,441)(157,448)(158,447)(159,446)
(160,445)(161,385)(162,386)(163,387)(164,388)(165,389)(166,390)(167,391)
(168,392)(169,394)(170,393)(171,396)(172,395)(173,398)(174,397)(175,400)
(176,399)(177,403)(178,404)(179,401)(180,402)(181,407)(182,408)(183,405)
(184,406)(185,412)(186,411)(187,410)(188,409)(189,416)(190,415)(191,414)
(192,413)(193,501)(194,502)(195,503)(196,504)(197,497)(198,498)(199,499)
(200,500)(201,510)(202,509)(203,512)(204,511)(205,506)(206,505)(207,508)
(208,507)(209,485)(210,486)(211,487)(212,488)(213,481)(214,482)(215,483)
(216,484)(217,494)(218,493)(219,496)(220,495)(221,490)(222,489)(223,492)
(224,491)(225,469)(226,470)(227,471)(228,472)(229,465)(230,466)(231,467)
(232,468)(233,478)(234,477)(235,480)(236,479)(237,474)(238,473)(239,476)
(240,475)(241,453)(242,454)(243,455)(244,456)(245,449)(246,450)(247,451)
(248,452)(249,462)(250,461)(251,464)(252,463)(253,458)(254,457)(255,460)
(256,459);
s2 := Sym(512)!(  1,129)(  2,130)(  3,131)(  4,132)(  5,133)(  6,134)(  7,135)
(  8,136)(  9,138)( 10,137)( 11,140)( 12,139)( 13,142)( 14,141)( 15,144)
( 16,143)( 17,149)( 18,150)( 19,151)( 20,152)( 21,145)( 22,146)( 23,147)
( 24,148)( 25,158)( 26,157)( 27,160)( 28,159)( 29,154)( 30,153)( 31,156)
( 32,155)( 33,169)( 34,170)( 35,171)( 36,172)( 37,173)( 38,174)( 39,175)
( 40,176)( 41,161)( 42,162)( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)
( 48,168)( 49,189)( 50,190)( 51,191)( 52,192)( 53,185)( 54,186)( 55,187)
( 56,188)( 57,181)( 58,182)( 59,183)( 60,184)( 61,177)( 62,178)( 63,179)
( 64,180)( 65,193)( 66,194)( 67,195)( 68,196)( 69,197)( 70,198)( 71,199)
( 72,200)( 73,202)( 74,201)( 75,204)( 76,203)( 77,206)( 78,205)( 79,208)
( 80,207)( 81,213)( 82,214)( 83,215)( 84,216)( 85,209)( 86,210)( 87,211)
( 88,212)( 89,222)( 90,221)( 91,224)( 92,223)( 93,218)( 94,217)( 95,220)
( 96,219)( 97,233)( 98,234)( 99,235)(100,236)(101,237)(102,238)(103,239)
(104,240)(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)
(112,232)(113,253)(114,254)(115,255)(116,256)(117,249)(118,250)(119,251)
(120,252)(121,245)(122,246)(123,247)(124,248)(125,241)(126,242)(127,243)
(128,244)(257,385)(258,386)(259,387)(260,388)(261,389)(262,390)(263,391)
(264,392)(265,394)(266,393)(267,396)(268,395)(269,398)(270,397)(271,400)
(272,399)(273,405)(274,406)(275,407)(276,408)(277,401)(278,402)(279,403)
(280,404)(281,414)(282,413)(283,416)(284,415)(285,410)(286,409)(287,412)
(288,411)(289,425)(290,426)(291,427)(292,428)(293,429)(294,430)(295,431)
(296,432)(297,417)(298,418)(299,419)(300,420)(301,421)(302,422)(303,423)
(304,424)(305,445)(306,446)(307,447)(308,448)(309,441)(310,442)(311,443)
(312,444)(313,437)(314,438)(315,439)(316,440)(317,433)(318,434)(319,435)
(320,436)(321,449)(322,450)(323,451)(324,452)(325,453)(326,454)(327,455)
(328,456)(329,458)(330,457)(331,460)(332,459)(333,462)(334,461)(335,464)
(336,463)(337,469)(338,470)(339,471)(340,472)(341,465)(342,466)(343,467)
(344,468)(345,478)(346,477)(347,480)(348,479)(349,474)(350,473)(351,476)
(352,475)(353,489)(354,490)(355,491)(356,492)(357,493)(358,494)(359,495)
(360,496)(361,481)(362,482)(363,483)(364,484)(365,485)(366,486)(367,487)
(368,488)(369,509)(370,510)(371,511)(372,512)(373,505)(374,506)(375,507)
(376,508)(377,501)(378,502)(379,503)(380,504)(381,497)(382,498)(383,499)
(384,500);
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*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*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 >; 
 
References : None.
to this polytope