#501235

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1113 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=i,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,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408431);