#506169

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	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

470 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)(17,18,22,26)(19,27,32,44)(20,33,35,38)(21,36,34,25)(23,37,40,30)(24,28,41,29)(31,48,47,43)(39,46,42,45);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19,28,45,21,31,20,30,22,32,29,46,34,47,35,37)(18,23,38,48,25,39,24,27,26,40,33,43,36,42,41,44);(2,6)(5,7)(11,15)(13,16);(2,6)(5,7)(17,19,29,46,21,31,35,37,22,32,28,45,34,47,20,30)(18,23,33,43,25,39,41,44,26,40,38,48,36,42,24,27)

Polycyclic presentation:

2^9|a^2=gi,b^2=ehi,d^2=efh,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^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, 413365);