#502902

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

843 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fh,d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di

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, 410098);