#501670

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

686 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)(11,19)(12,24)(13,16)(14,27)(15,21)(17,20)(18,25)(22,31)(23,38)(26,39)(28,37)(29,40)(30,35)(32,34)(33,36);(5,6)(7,8)(9,11,20,34,14,23,26,37,13,22,35,40,25,36,12,21)(10,15,24,33,18,29,30,31,16,28,39,38,27,32,17,19);(2,4)(6,8)(10,16)(15,28)(17,30)(18,27)(19,31)(24,39)(29,32)(33,38);(9,11,20,34,14,23,26,37,13,22,35,40,25,36,12,21)(10,15,24,33,18,29,30,31,16,28,39,38,27,32,17,19)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408866);