#506131

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, 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

414 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)(17,18,24,31)(19,33,36,48)(20,28,37,46)(21,27,40,44)(22,30,39,47)(23,29,41,43)(25,42,34,32)(26,38,45,35);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,20,25,21,39,38,24,36,41,37,34,40,22,35)(18,26,30,44,32,46,43,48,31,45,47,27,42,28,29,33);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20,22,19,34,35,41,40,24,37,39,36,25,38,23,21)(18,27,29,45,42,48,47,46,31,44,43,26,32,33,30,28);(2,7)(6,8)(17,21,23,38,25,36,39,37,24,40,41,35,34,19,22,20)(18,28,30,33,32,26,43,44,31,46,47,48,42,45,29,27)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,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^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, 413327);