#7624644

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624184
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 ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1318 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:

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

Polycyclic presentation:

2^9|c^2=i,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531840);