#507353

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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 ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

482 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,5)(3,7,8,6)(9,10,12,14)(11,15,16,13)(17,18,20,24)(19,27,30,23)(21,25,28,35)(22,26,33,39)(29,41,46,36)(31,42,44,37)(32,43,47,38)(34,40,45,48);(2,5)(6,7)(9,11,12,16)(10,13,14,15)(17,19,28,44,33,47,34,29,20,30,21,31,22,32,45,46)(18,23,35,42,39,43,40,36,24,27,25,37,26,38,48,41);(10,14)(13,15)(19,29)(21,34)(22,33)(23,36)(25,40)(26,39)(27,41)(28,45)(30,46)(31,32)(35,48)(37,38)(42,43)(44,47);(1,3,4,8)(2,6,5,7)(10,14)(13,15)

Polycyclic presentation:

2^9|a^2=e,b^2=fgi,d^2=efi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,g^c=gh,h^c=hi

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, 414549);