#7623379

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

1632 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,17)(13,26)(14,24)(15,19)(16,22)(18,28)(21,37)(23,38)(25,35)(27,31)(29,36)(30,34)(32,40)(33,39);(1,3,4,8)(2,5,6,7);(3,8)(5,7)(9,11,15,23)(10,16,19,33)(12,21,14,30)(13,27,29,40)(17,31,24,32)(18,34,35,37)(20,36,38,26)(22,25,39,28);(2,6)(5,7)(9,10)(11,16)(12,24)(13,18)(14,17)(15,19)(20,22)(21,32)(23,33)(25,36)(26,28)(27,34)(29,35)(30,31)(37,40)(38,39);(9,12,15,14)(10,17,19,24)(11,21,23,30)(13,25,29,28)(16,31,33,32)(18,26,35,36)(20,37,38,34)(22,27,39,40)

Polycyclic presentation:

2^9|b^2=f,c^2=i,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=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, 7530575);