#506090

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

394 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)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18,24,31)(19,33,37,48)(20,38,39,28)(21,27,40,47)(22,41,35,30)(23,43,42,29)(25,44,34,32)(26,45,46,36);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,20,25,21,22,36,24,37,42,39,34,40,35,45)(18,26,30,27,32,38,29,33,31,46,41,47,44,28,43,48);(11,15)(13,16)(17,20,35,19,34,45,42,40,24,39,22,37,25,36,23,21)(18,27,43,26,44,48,41,28,31,47,29,46,32,33,30,38);(1,3,4,8)(2,6,5,7)(17,21,23,36,25,37,22,39,24,40,42,45,34,19,35,20)(18,28,30,48,32,26,29,27,31,38,41,33,44,46,43,47)

Polycyclic presentation:

2^9|a^2=g,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, 413286);