#506114

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

451 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:

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

Polycyclic presentation:

2^9|a^2=efgi,b^2=f,c^2=fh,d^2=ehi,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, 413310);