#504282

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411478);