#7624981

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624812
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#62418	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532177);