#502124

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

812 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)(9,10,14,20,12,28,30,31,16,21,13,19,15,18,29,32)(11,17,25,36,23,39,33,38,27,37,24,35,26,34,40,22);(2,4)(5,6)(7,8)(9,11,13,24,12,23,29,40,16,27,14,25,15,26,30,33)(10,17,19,35,28,39,32,22,21,37,20,36,18,34,31,38);(6,8)(9,10,16,21)(11,22,27,38)(12,18,15,28)(13,31,14,32)(17,33,37,40)(19,29,20,30)(23,35,26,36)(24,39,25,34);(2,4)(9,12,16,15)(10,18,21,28)(11,23,27,26)(13,29,14,30)(17,34,37,39)(19,31,20,32)(22,35,38,36)(24,40,25,33)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,g^c=gh,h^c=hi

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, 409320);