#7623271

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622979
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
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1390 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530467);