#72234408

order	8192
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆² · Schur multiplier (H₂) = C₈²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₃₂	C₁₆²
commutator	C₁₆	C₂×C₁₆²
Frattini, cosocle	#72234239	C₂³
socle	C₂	#72234269
prime radical	C₂³	#72210687
LCS	G ▷ C₁₆ ▷ C₁
UCS	C₁ ◁ C₃₂ ◁ G
LECS	G ▷ #72234239 ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1276 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 512
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 512 points:

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

Polycyclic presentation:

2^13|a^2=d,b^2=e,c^2=fjl,d^2=g,e^2=h,f^2=ilm,g^2=j,h^2=k,i^2=l,j^2=l,k^2=m,l^2=m,b^a=bf,c^b=cl,d^b=di,e^a=eilm,e^c=em,e^d=elm,g^b=gl,g^e=gm,h^a=hlm,h^d=hm,j^b=jm,k^a=km

Conjugacy classes & character table

752 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72234409

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l", "m");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6*F.10*F.12);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9*F.12*F.13);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.11);;
gap> SetPower(coll, 9, F.12);;
gap> SetPower(coll, 10, F.12);;
gap> SetPower(coll, 11, F.13);;
gap> SetPower(coll, 12, F.13);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.12);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.12*F.13);;
gap> SetConjugate(coll, 5, 3, F.5*F.13);;
gap> SetConjugate(coll, 5, 4, F.5*F.12*F.13);;
gap> SetConjugate(coll, 7, 2, F.7*F.12);;
gap> SetConjugate(coll, 7, 5, F.7*F.13);;
gap> SetConjugate(coll, 8, 1, F.8*F.12*F.13);;
gap> SetConjugate(coll, 8, 4, F.8*F.13);;
gap> SetConjugate(coll, 10, 2, F.10*F.13);;
gap> SetConjugate(coll, 11, 1, F.11*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);