#512331

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

578 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)(13,14,20,24)(15,25,26,40)(16,18,32,34)(17,31,35,22)(19,30,33,23)(21,36,37,27)(28,39,43,41)(29,38,44,42);(5,7,8,11)(6,9,10,12)(13,15,18,28,19,29,17,27,20,26,34,43,33,44,35,36)(14,21,22,38,23,39,16,25,24,37,31,42,30,41,32,40);(6,10)(9,12)(15,26)(21,37)(25,40)(27,36)(28,43)(29,44)(38,42)(39,41);(2,4)(13,16,17,30,33,22,18,24,20,32,35,23,19,31,34,14)(15,25,27,41,44,38,28,37,26,40,36,39,29,42,43,21)

Polycyclic presentation:

2^9|a^2=i,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, 419527);