#503753

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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

532 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410949);