#506156

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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

484 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18,21,24)(19,27,29,42)(20,32,34,47)(22,35,33,26)(23,36,37,30)(25,40,39,28)(31,46,45,43)(38,44,41,48);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19,28,44,22,31,34,36,21,29,40,48,33,45,20,30)(18,23,32,43,26,38,39,42,24,37,47,46,35,41,25,27);(2,6)(5,7)(11,15)(13,16)(18,24)(19,29)(25,39)(26,35)(30,36)(31,45)(32,47)(44,48);(2,6)(5,7)(17,19,28,44,22,31,34,36,21,29,40,48,33,45,20,30)(18,23,32,43,26,38,39,42,24,37,47,46,35,41,25,27)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=efhi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^c=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, 413352);