#506158

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

467 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)(17,18,22,27)(19,29,20,34)(21,37,39,41)(23,40,38,28)(24,35,25,32)(26,30,44,31)(33,48,36,45)(42,47,43,46);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19,30,46,23,33,21,32,22,20,31,47,38,36,39,35)(18,24,41,48,28,42,26,29,27,25,37,45,40,43,44,34);(2,6)(5,7)(11,15)(13,16)(19,20)(24,25)(29,34)(32,35)(33,36)(42,43)(45,48)(46,47);(2,6)(5,7)(17,20,30,47,23,36,21,35,22,19,31,46,38,33,39,32)(18,25,41,45,28,43,26,34,27,24,37,48,40,42,44,29)

Polycyclic presentation:

2^9|a^2=g,b^2=ehi,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, 413354);