#7623384

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623326
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

1644 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530580);