#6808931

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1713 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,27,32)(23,28,36,46)(24,30,38,48)(25,31,26,41)(29,40,49,37)(33,50,34,43)(35,44,39,45)(42,51,47,52);(2,3)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23)(22,28)(24,34)(25,39)(26,35)(27,36)(29,47)(30,43)(31,45)(32,46)(33,38)(37,51)(40,52)(41,44)(42,49)(48,50);(1,2,4,3)(21,24)(22,29)(23,33)(25,37)(26,40)(27,38)(28,42)(30,41)(31,48)(32,49)(34,36)(35,51)(39,52)(43,45)(44,50)(46,47);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,24)(22,30)(23,34)(25,40)(26,37)(27,38)(28,43)(29,41)(31,49)(32,48)(33,36)(35,51)(39,52)(42,45)(44,47)(46,50);(1,3,4,2)(7,15)(10,13)(14,20)(18,19)(21,25,27,26)(22,31,32,41)(23,35,36,39)(24,37,38,40)(28,44,46,45)(29,48,49,30)(33,51,34,52)(42,50,47,43)

Polycyclic presentation:

2^9|a^2=fi,c^2=hi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6716127);