#507485

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

564 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)(11,13,15,16)(17,18,20,24)(19,27,29,41)(21,34,32,47)(22,35,33,26)(23,36,37,30)(25,40,39,28)(31,46,45,43)(38,44,42,48);(2,5)(6,7)(9,11,12,15)(10,13,14,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,34)(29,37)(31,38)(32,39)(33,35)(36,41)(40,47)(42,45)(43,44)(46,48);(11,15)(13,16)(17,19,28,44,22,31,32,36,20,29,40,48,33,45,21,30)(18,23,34,43,26,38,39,41,24,37,47,46,35,42,25,27);(1,3,4,8)(2,6,5,7)(18,24)(23,37)(25,39)(26,35)(27,41)(34,47)(38,42)(43,46)

Polycyclic presentation:

2^9|a^2=eg,b^2=g,c^2=fh,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,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, 414681);