#506124

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

427 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18,24,31)(19,33,36,48)(20,38,39,28)(21,41,37,27)(22,30,43,47)(23,29,42,45)(25,44,34,32)(26,40,46,35);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,20,25,37,43,40,24,36,42,39,34,21,22,35)(18,26,30,41,32,28,45,48,31,46,47,27,44,38,29,33);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20,22,19,34,35,42,21,24,39,43,36,25,40,23,37)(18,27,29,46,44,48,47,28,31,41,45,26,32,33,30,38);(2,7)(6,8)(17,21,42,40,25,19,22,39,24,37,23,35,34,36,43,20)(18,28,47,33,32,46,29,41,31,38,30,48,44,26,45,27)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=e,d^2=fi,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^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, 413320);