#506129

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, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

410 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ehi,c^2=e,d^2=f,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, 413325);