#512322

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

599 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)(21,36)(22,23)(24,35)(26,39)(27,41)(28,40)(32,37)(33,34)(42,44);(5,7,8,11)(6,9,10,12)(13,14,17,22,18,23,16,20,19,24,29,34,30,33,31,35)(15,21,25,37,28,38,27,32,26,36,39,44,41,43,40,42);(6,10)(9,12)(15,26)(21,36)(25,39)(27,40)(28,41)(32,42)(37,44)(38,43);(2,4)(13,15,16,27,30,41,17,25,19,26,31,40,18,28,29,39)(14,21,20,32,33,43,22,37,24,36,35,42,23,38,34,44)

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,d^c=di

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, 419518);