Questions?
See the FAQ
or other info.

Polytope of Type {16,16}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {16,16}*512b
Also Known As : {16,16|2}. if this polytope has another name.
Group : SmallGroup(512,30471)
Rank : 3
Schlafli Type : {16,16}
Number of vertices, edges, etc : 16, 128, 16
Order of s0s1s2 : 16
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Dual
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 : {8,16}*256d, {16,8}*256d
   4-fold quotients : {8,8}*128b, {4,16}*128a, {16,4}*128a
   8-fold quotients : {4,8}*64a, {8,4}*64a, {2,16}*64, {16,2}*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, 70)(  6, 69)(  7, 72)(  8, 71)
(  9, 74)( 10, 73)( 11, 76)( 12, 75)( 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, 94)( 26, 93)( 27, 96)( 28, 95)( 29, 90)( 30, 89)( 31, 92)( 32, 91)
( 33,101)( 34,102)( 35,103)( 36,104)( 37, 97)( 38, 98)( 39, 99)( 40,100)
( 41,110)( 42,109)( 43,112)( 44,111)( 45,106)( 46,105)( 47,108)( 48,107)
( 49,114)( 50,113)( 51,116)( 52,115)( 53,117)( 54,118)( 55,119)( 56,120)
( 57,121)( 58,122)( 59,123)( 60,124)( 61,126)( 62,125)( 63,128)( 64,127)
(129,193)(130,194)(131,195)(132,196)(133,198)(134,197)(135,200)(136,199)
(137,202)(138,201)(139,204)(140,203)(141,205)(142,206)(143,207)(144,208)
(145,213)(146,214)(147,215)(148,216)(149,209)(150,210)(151,211)(152,212)
(153,222)(154,221)(155,224)(156,223)(157,218)(158,217)(159,220)(160,219)
(161,229)(162,230)(163,231)(164,232)(165,225)(166,226)(167,227)(168,228)
(169,238)(170,237)(171,240)(172,239)(173,234)(174,233)(175,236)(176,235)
(177,242)(178,241)(179,244)(180,243)(181,245)(182,246)(183,247)(184,248)
(185,249)(186,250)(187,251)(188,252)(189,254)(190,253)(191,256)(192,255)
(257,321)(258,322)(259,323)(260,324)(261,326)(262,325)(263,328)(264,327)
(265,330)(266,329)(267,332)(268,331)(269,333)(270,334)(271,335)(272,336)
(273,341)(274,342)(275,343)(276,344)(277,337)(278,338)(279,339)(280,340)
(281,350)(282,349)(283,352)(284,351)(285,346)(286,345)(287,348)(288,347)
(289,357)(290,358)(291,359)(292,360)(293,353)(294,354)(295,355)(296,356)
(297,366)(298,365)(299,368)(300,367)(301,362)(302,361)(303,364)(304,363)
(305,370)(306,369)(307,372)(308,371)(309,373)(310,374)(311,375)(312,376)
(313,377)(314,378)(315,379)(316,380)(317,382)(318,381)(319,384)(320,383)
(385,449)(386,450)(387,451)(388,452)(389,454)(390,453)(391,456)(392,455)
(393,458)(394,457)(395,460)(396,459)(397,461)(398,462)(399,463)(400,464)
(401,469)(402,470)(403,471)(404,472)(405,465)(406,466)(407,467)(408,468)
(409,478)(410,477)(411,480)(412,479)(413,474)(414,473)(415,476)(416,475)
(417,485)(418,486)(419,487)(420,488)(421,481)(422,482)(423,483)(424,484)
(425,494)(426,493)(427,496)(428,495)(429,490)(430,489)(431,492)(432,491)
(433,498)(434,497)(435,500)(436,499)(437,501)(438,502)(439,503)(440,504)
(441,505)(442,506)(443,507)(444,508)(445,510)(446,509)(447,512)(448,511);;
s1 := (  1,257)(  2,258)(  3,259)(  4,260)(  5,262)(  6,261)(  7,264)(  8,263)
(  9,267)( 10,268)( 11,265)( 12,266)( 13,272)( 14,271)( 15,270)( 16,269)
( 17,277)( 18,278)( 19,279)( 20,280)( 21,273)( 22,274)( 23,275)( 24,276)
( 25,287)( 26,288)( 27,285)( 28,286)( 29,283)( 30,284)( 31,281)( 32,282)
( 33,297)( 34,298)( 35,299)( 36,300)( 37,302)( 38,301)( 39,304)( 40,303)
( 41,289)( 42,290)( 43,291)( 44,292)( 45,294)( 46,293)( 47,296)( 48,295)
( 49,317)( 50,318)( 51,319)( 52,320)( 53,313)( 54,314)( 55,315)( 56,316)
( 57,309)( 58,310)( 59,311)( 60,312)( 61,305)( 62,306)( 63,307)( 64,308)
( 65,337)( 66,338)( 67,339)( 68,340)( 69,342)( 70,341)( 71,344)( 72,343)
( 73,347)( 74,348)( 75,345)( 76,346)( 77,352)( 78,351)( 79,350)( 80,349)
( 81,321)( 82,322)( 83,323)( 84,324)( 85,326)( 86,325)( 87,328)( 88,327)
( 89,331)( 90,332)( 91,329)( 92,330)( 93,336)( 94,335)( 95,334)( 96,333)
( 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)
(129,417)(130,418)(131,419)(132,420)(133,422)(134,421)(135,424)(136,423)
(137,427)(138,428)(139,425)(140,426)(141,432)(142,431)(143,430)(144,429)
(145,437)(146,438)(147,439)(148,440)(149,433)(150,434)(151,435)(152,436)
(153,447)(154,448)(155,445)(156,446)(157,443)(158,444)(159,441)(160,442)
(161,385)(162,386)(163,387)(164,388)(165,390)(166,389)(167,392)(168,391)
(169,395)(170,396)(171,393)(172,394)(173,400)(174,399)(175,398)(176,397)
(177,405)(178,406)(179,407)(180,408)(181,401)(182,402)(183,403)(184,404)
(185,415)(186,416)(187,413)(188,414)(189,411)(190,412)(191,409)(192,410)
(193,501)(194,502)(195,503)(196,504)(197,497)(198,498)(199,499)(200,500)
(201,511)(202,512)(203,509)(204,510)(205,507)(206,508)(207,505)(208,506)
(209,485)(210,486)(211,487)(212,488)(213,481)(214,482)(215,483)(216,484)
(217,495)(218,496)(219,493)(220,494)(221,491)(222,492)(223,489)(224,490)
(225,469)(226,470)(227,471)(228,472)(229,465)(230,466)(231,467)(232,468)
(233,479)(234,480)(235,477)(236,478)(237,475)(238,476)(239,473)(240,474)
(241,453)(242,454)(243,455)(244,456)(245,449)(246,450)(247,451)(248,452)
(249,463)(250,464)(251,461)(252,462)(253,459)(254,460)(255,457)(256,458);;
s2 := (  1,193)(  2,194)(  3,195)(  4,196)(  5,197)(  6,198)(  7,199)(  8,200)
(  9,204)( 10,203)( 11,202)( 12,201)( 13,208)( 14,207)( 15,206)( 16,205)
( 17,209)( 18,210)( 19,211)( 20,212)( 21,213)( 22,214)( 23,215)( 24,216)
( 25,220)( 26,219)( 27,218)( 28,217)( 29,224)( 30,223)( 31,222)( 32,221)
( 33,238)( 34,237)( 35,240)( 36,239)( 37,233)( 38,234)( 39,235)( 40,236)
( 41,229)( 42,230)( 43,231)( 44,232)( 45,226)( 46,225)( 47,228)( 48,227)
( 49,254)( 50,253)( 51,256)( 52,255)( 53,249)( 54,250)( 55,251)( 56,252)
( 57,245)( 58,246)( 59,247)( 60,248)( 61,242)( 62,241)( 63,244)( 64,243)
( 65,129)( 66,130)( 67,131)( 68,132)( 69,133)( 70,134)( 71,135)( 72,136)
( 73,140)( 74,139)( 75,138)( 76,137)( 77,144)( 78,143)( 79,142)( 80,141)
( 81,145)( 82,146)( 83,147)( 84,148)( 85,149)( 86,150)( 87,151)( 88,152)
( 89,156)( 90,155)( 91,154)( 92,153)( 93,160)( 94,159)( 95,158)( 96,157)
( 97,174)( 98,173)( 99,176)(100,175)(101,169)(102,170)(103,171)(104,172)
(105,165)(106,166)(107,167)(108,168)(109,162)(110,161)(111,164)(112,163)
(113,190)(114,189)(115,192)(116,191)(117,185)(118,186)(119,187)(120,188)
(121,181)(122,182)(123,183)(124,184)(125,178)(126,177)(127,180)(128,179)
(257,449)(258,450)(259,451)(260,452)(261,453)(262,454)(263,455)(264,456)
(265,460)(266,459)(267,458)(268,457)(269,464)(270,463)(271,462)(272,461)
(273,465)(274,466)(275,467)(276,468)(277,469)(278,470)(279,471)(280,472)
(281,476)(282,475)(283,474)(284,473)(285,480)(286,479)(287,478)(288,477)
(289,494)(290,493)(291,496)(292,495)(293,489)(294,490)(295,491)(296,492)
(297,485)(298,486)(299,487)(300,488)(301,482)(302,481)(303,484)(304,483)
(305,510)(306,509)(307,512)(308,511)(309,505)(310,506)(311,507)(312,508)
(313,501)(314,502)(315,503)(316,504)(317,498)(318,497)(319,500)(320,499)
(321,385)(322,386)(323,387)(324,388)(325,389)(326,390)(327,391)(328,392)
(329,396)(330,395)(331,394)(332,393)(333,400)(334,399)(335,398)(336,397)
(337,401)(338,402)(339,403)(340,404)(341,405)(342,406)(343,407)(344,408)
(345,412)(346,411)(347,410)(348,409)(349,416)(350,415)(351,414)(352,413)
(353,430)(354,429)(355,432)(356,431)(357,425)(358,426)(359,427)(360,428)
(361,421)(362,422)(363,423)(364,424)(365,418)(366,417)(367,420)(368,419)
(369,446)(370,445)(371,448)(372,447)(373,441)(374,442)(375,443)(376,444)
(377,437)(378,438)(379,439)(380,440)(381,434)(382,433)(383,436)(384,435);;
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*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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 ];;
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, 74)( 10, 73)( 11, 76)( 12, 75)( 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, 94)( 26, 93)( 27, 96)( 28, 95)( 29, 90)( 30, 89)( 31, 92)
( 32, 91)( 33,101)( 34,102)( 35,103)( 36,104)( 37, 97)( 38, 98)( 39, 99)
( 40,100)( 41,110)( 42,109)( 43,112)( 44,111)( 45,106)( 46,105)( 47,108)
( 48,107)( 49,114)( 50,113)( 51,116)( 52,115)( 53,117)( 54,118)( 55,119)
( 56,120)( 57,121)( 58,122)( 59,123)( 60,124)( 61,126)( 62,125)( 63,128)
( 64,127)(129,193)(130,194)(131,195)(132,196)(133,198)(134,197)(135,200)
(136,199)(137,202)(138,201)(139,204)(140,203)(141,205)(142,206)(143,207)
(144,208)(145,213)(146,214)(147,215)(148,216)(149,209)(150,210)(151,211)
(152,212)(153,222)(154,221)(155,224)(156,223)(157,218)(158,217)(159,220)
(160,219)(161,229)(162,230)(163,231)(164,232)(165,225)(166,226)(167,227)
(168,228)(169,238)(170,237)(171,240)(172,239)(173,234)(174,233)(175,236)
(176,235)(177,242)(178,241)(179,244)(180,243)(181,245)(182,246)(183,247)
(184,248)(185,249)(186,250)(187,251)(188,252)(189,254)(190,253)(191,256)
(192,255)(257,321)(258,322)(259,323)(260,324)(261,326)(262,325)(263,328)
(264,327)(265,330)(266,329)(267,332)(268,331)(269,333)(270,334)(271,335)
(272,336)(273,341)(274,342)(275,343)(276,344)(277,337)(278,338)(279,339)
(280,340)(281,350)(282,349)(283,352)(284,351)(285,346)(286,345)(287,348)
(288,347)(289,357)(290,358)(291,359)(292,360)(293,353)(294,354)(295,355)
(296,356)(297,366)(298,365)(299,368)(300,367)(301,362)(302,361)(303,364)
(304,363)(305,370)(306,369)(307,372)(308,371)(309,373)(310,374)(311,375)
(312,376)(313,377)(314,378)(315,379)(316,380)(317,382)(318,381)(319,384)
(320,383)(385,449)(386,450)(387,451)(388,452)(389,454)(390,453)(391,456)
(392,455)(393,458)(394,457)(395,460)(396,459)(397,461)(398,462)(399,463)
(400,464)(401,469)(402,470)(403,471)(404,472)(405,465)(406,466)(407,467)
(408,468)(409,478)(410,477)(411,480)(412,479)(413,474)(414,473)(415,476)
(416,475)(417,485)(418,486)(419,487)(420,488)(421,481)(422,482)(423,483)
(424,484)(425,494)(426,493)(427,496)(428,495)(429,490)(430,489)(431,492)
(432,491)(433,498)(434,497)(435,500)(436,499)(437,501)(438,502)(439,503)
(440,504)(441,505)(442,506)(443,507)(444,508)(445,510)(446,509)(447,512)
(448,511);
s1 := Sym(512)!(  1,257)(  2,258)(  3,259)(  4,260)(  5,262)(  6,261)(  7,264)
(  8,263)(  9,267)( 10,268)( 11,265)( 12,266)( 13,272)( 14,271)( 15,270)
( 16,269)( 17,277)( 18,278)( 19,279)( 20,280)( 21,273)( 22,274)( 23,275)
( 24,276)( 25,287)( 26,288)( 27,285)( 28,286)( 29,283)( 30,284)( 31,281)
( 32,282)( 33,297)( 34,298)( 35,299)( 36,300)( 37,302)( 38,301)( 39,304)
( 40,303)( 41,289)( 42,290)( 43,291)( 44,292)( 45,294)( 46,293)( 47,296)
( 48,295)( 49,317)( 50,318)( 51,319)( 52,320)( 53,313)( 54,314)( 55,315)
( 56,316)( 57,309)( 58,310)( 59,311)( 60,312)( 61,305)( 62,306)( 63,307)
( 64,308)( 65,337)( 66,338)( 67,339)( 68,340)( 69,342)( 70,341)( 71,344)
( 72,343)( 73,347)( 74,348)( 75,345)( 76,346)( 77,352)( 78,351)( 79,350)
( 80,349)( 81,321)( 82,322)( 83,323)( 84,324)( 85,326)( 86,325)( 87,328)
( 88,327)( 89,331)( 90,332)( 91,329)( 92,330)( 93,336)( 94,335)( 95,334)
( 96,333)( 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)(129,417)(130,418)(131,419)(132,420)(133,422)(134,421)(135,424)
(136,423)(137,427)(138,428)(139,425)(140,426)(141,432)(142,431)(143,430)
(144,429)(145,437)(146,438)(147,439)(148,440)(149,433)(150,434)(151,435)
(152,436)(153,447)(154,448)(155,445)(156,446)(157,443)(158,444)(159,441)
(160,442)(161,385)(162,386)(163,387)(164,388)(165,390)(166,389)(167,392)
(168,391)(169,395)(170,396)(171,393)(172,394)(173,400)(174,399)(175,398)
(176,397)(177,405)(178,406)(179,407)(180,408)(181,401)(182,402)(183,403)
(184,404)(185,415)(186,416)(187,413)(188,414)(189,411)(190,412)(191,409)
(192,410)(193,501)(194,502)(195,503)(196,504)(197,497)(198,498)(199,499)
(200,500)(201,511)(202,512)(203,509)(204,510)(205,507)(206,508)(207,505)
(208,506)(209,485)(210,486)(211,487)(212,488)(213,481)(214,482)(215,483)
(216,484)(217,495)(218,496)(219,493)(220,494)(221,491)(222,492)(223,489)
(224,490)(225,469)(226,470)(227,471)(228,472)(229,465)(230,466)(231,467)
(232,468)(233,479)(234,480)(235,477)(236,478)(237,475)(238,476)(239,473)
(240,474)(241,453)(242,454)(243,455)(244,456)(245,449)(246,450)(247,451)
(248,452)(249,463)(250,464)(251,461)(252,462)(253,459)(254,460)(255,457)
(256,458);
s2 := Sym(512)!(  1,193)(  2,194)(  3,195)(  4,196)(  5,197)(  6,198)(  7,199)
(  8,200)(  9,204)( 10,203)( 11,202)( 12,201)( 13,208)( 14,207)( 15,206)
( 16,205)( 17,209)( 18,210)( 19,211)( 20,212)( 21,213)( 22,214)( 23,215)
( 24,216)( 25,220)( 26,219)( 27,218)( 28,217)( 29,224)( 30,223)( 31,222)
( 32,221)( 33,238)( 34,237)( 35,240)( 36,239)( 37,233)( 38,234)( 39,235)
( 40,236)( 41,229)( 42,230)( 43,231)( 44,232)( 45,226)( 46,225)( 47,228)
( 48,227)( 49,254)( 50,253)( 51,256)( 52,255)( 53,249)( 54,250)( 55,251)
( 56,252)( 57,245)( 58,246)( 59,247)( 60,248)( 61,242)( 62,241)( 63,244)
( 64,243)( 65,129)( 66,130)( 67,131)( 68,132)( 69,133)( 70,134)( 71,135)
( 72,136)( 73,140)( 74,139)( 75,138)( 76,137)( 77,144)( 78,143)( 79,142)
( 80,141)( 81,145)( 82,146)( 83,147)( 84,148)( 85,149)( 86,150)( 87,151)
( 88,152)( 89,156)( 90,155)( 91,154)( 92,153)( 93,160)( 94,159)( 95,158)
( 96,157)( 97,174)( 98,173)( 99,176)(100,175)(101,169)(102,170)(103,171)
(104,172)(105,165)(106,166)(107,167)(108,168)(109,162)(110,161)(111,164)
(112,163)(113,190)(114,189)(115,192)(116,191)(117,185)(118,186)(119,187)
(120,188)(121,181)(122,182)(123,183)(124,184)(125,178)(126,177)(127,180)
(128,179)(257,449)(258,450)(259,451)(260,452)(261,453)(262,454)(263,455)
(264,456)(265,460)(266,459)(267,458)(268,457)(269,464)(270,463)(271,462)
(272,461)(273,465)(274,466)(275,467)(276,468)(277,469)(278,470)(279,471)
(280,472)(281,476)(282,475)(283,474)(284,473)(285,480)(286,479)(287,478)
(288,477)(289,494)(290,493)(291,496)(292,495)(293,489)(294,490)(295,491)
(296,492)(297,485)(298,486)(299,487)(300,488)(301,482)(302,481)(303,484)
(304,483)(305,510)(306,509)(307,512)(308,511)(309,505)(310,506)(311,507)
(312,508)(313,501)(314,502)(315,503)(316,504)(317,498)(318,497)(319,500)
(320,499)(321,385)(322,386)(323,387)(324,388)(325,389)(326,390)(327,391)
(328,392)(329,396)(330,395)(331,394)(332,393)(333,400)(334,399)(335,398)
(336,397)(337,401)(338,402)(339,403)(340,404)(341,405)(342,406)(343,407)
(344,408)(345,412)(346,411)(347,410)(348,409)(349,416)(350,415)(351,414)
(352,413)(353,430)(354,429)(355,432)(356,431)(357,425)(358,426)(359,427)
(360,428)(361,421)(362,422)(363,423)(364,424)(365,418)(366,417)(367,420)
(368,419)(369,446)(370,445)(371,448)(372,447)(373,441)(374,442)(375,443)
(376,444)(377,437)(378,438)(379,439)(380,440)(381,434)(382,433)(383,436)
(384,435);
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*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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 >; 
 
References : None.
to this polytope