#506174

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	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

480 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)(19,27,29,43)(20,33,35,38)(22,36,34,26)(23,37,39,31)(25,28,41,30)(32,48,46,44)(40,47,42,45);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19,28,45,22,32,20,31,21,29,30,47,34,46,35,37)(18,23,38,48,26,40,25,27,24,39,33,44,36,42,41,43);(2,6)(5,7)(11,15)(13,16)(18,24)(19,29)(25,41)(26,36)(31,37)(32,46)(33,38)(45,47);(2,6)(5,7)(17,19,30,47,22,32,35,37,21,29,28,45,34,46,20,31)(18,23,33,44,26,40,41,43,24,39,38,48,36,42,25,27)

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^b=di,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, 413370);