#501841

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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 ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

683 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409037);