#506132

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

423 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18,24,31)(19,33,37,48)(20,28,40,47)(21,41,38,27)(22,43,35,30)(23,44,34,29)(25,45,39,32)(26,42,46,36);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,34,40,25,38,35,42,24,37,23,20,39,21,22,36)(18,26,43,41,32,28,44,48,31,46,30,27,45,47,29,33);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20,22,37,39,42,23,38,24,40,35,19,25,36,34,21)(18,27,29,46,45,48,30,28,31,41,44,26,32,33,43,47);(2,7)(6,8)(17,21,34,36,25,19,35,40,24,38,23,42,39,37,22,20)(18,28,43,48,32,46,44,27,31,47,30,33,45,26,29,41)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=e,d^2=fi,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, 413328);