#7623375

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623323
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

1664 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)(10,15)(12,22)(13,24)(16,28)(17,32)(18,29)(20,35)(21,34)(23,26)(25,27)(30,39)(31,40)(33,38)(36,37);(1,3,4,8)(2,5,6,7);(3,8)(5,7)(9,10,14,18)(11,16,19,30)(12,23,25,32)(13,26,24,17)(15,27,29,22)(20,36,33,40)(21,37,34,31)(28,38,39,35);(2,6)(5,7)(9,11)(10,16)(12,20)(13,21)(14,19)(15,28)(17,31)(18,30)(22,35)(23,36)(24,34)(25,33)(26,37)(27,38)(29,39)(32,40);(11,19)(16,30)(20,33)(21,34)(28,39)(31,37)(35,38)(36,40)

Polycyclic presentation:

2^9|b^2=f,c^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=cf,g^c=gh,h^c=hi,e^d=ei

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, 7530571);