#7623762

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623626
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 ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1513 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,d^b=df,e^b=ei,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, 7530958);