#6467297

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1970 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=hi,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374493);