#507486

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

508 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=egi,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, 414682);