#500837

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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

1011 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)(11,21)(12,26)(13,29)(14,19)(15,32)(16,24)(17,28)(18,22)(20,30)(23,37)(25,40)(27,34)(31,36)(33,39)(35,38);(5,6)(7,8)(9,11,22,38,15,25,31,39,14,23,27,28,30,12,13,24)(10,16,29,26,20,17,34,37,19,33,36,40,32,35,18,21);(2,4)(6,8)(11,23)(12,25)(16,33)(17,35)(21,37)(24,39)(26,40)(28,38);(9,12,27,39,15,11,13,28,14,25,22,24,30,23,31,38)(10,17,36,21,20,33,18,26,19,35,29,37,32,16,34,40)

Polycyclic presentation:

2^9|b^2=eh,d^2=ehi,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^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, 408033);