#6683111

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	C₂×#3061	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1996 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6590307);