#506092

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

398 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)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18,25,32)(19,33,35,48)(20,28,38,45)(21,27,39,44)(22,30,36,47)(23,29,41,43)(24,42,40,31)(26,37,46,34);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,20,24,21,36,37,25,35,41,38,40,39,22,34)(18,26,30,44,31,45,43,48,32,46,47,27,42,28,29,33);(11,15)(13,16)(17,20,36,35,40,34,23,21,25,38,22,19,24,37,41,39)(18,27,43,26,42,48,30,28,32,44,29,46,31,33,47,45);(1,3,4,8)(2,6,5,7)(17,21,23,37,24,35,36,38,25,39,41,34,40,19,22,20)(18,28,30,33,31,26,43,44,32,45,47,48,42,46,29,27)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=fh,d^2=eh,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,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, 413288);