#506171

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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

480 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18,22,25)(19,29,20,33)(21,37,39,43)(23,40,38,28)(24,35,26,31)(27,34,42,30)(32,48,36,45)(41,46,44,47);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19,30,46,23,32,39,35,22,20,34,47,38,36,21,31)(18,24,37,45,28,41,42,33,25,26,43,48,40,44,27,29);(2,6)(5,7)(11,15)(13,16)(18,25)(19,20)(27,42)(28,40)(31,35)(32,36)(37,43)(46,47);(2,6)(5,7)(17,20,34,46,23,36,21,35,22,19,30,47,38,32,39,31)(18,26,43,45,28,44,27,33,25,24,37,48,40,41,42,29)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=efh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,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, 413367);