#512329

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

Polycyclic presentation:

2^9|b^2=efh,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, 419525);