#512330

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2218	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=efhi,d^2=e,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=ghi,h^a=hi,c^b=ci

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419526);