#506111

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

434 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:

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

Polycyclic presentation:

2^9|a^2=efg,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, 413307);