#7624854

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624824
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

1337 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)(11,19)(12,20)(13,23)(15,25)(17,26)(18,30)(21,35)(22,33)(24,34)(27,36)(28,40)(29,39)(31,32)(37,38);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,25)(20,26)(21,28)(22,27)(23,30)(24,29)(31,37)(32,38)(33,36)(34,39)(35,40);(9,11,20,32,13,21,24,33,14,22,34,35,23,31,12,19)(10,15,26,38,18,28,29,36,16,27,39,40,30,37,17,25);(3,8)(5,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,25)(20,26)(21,28)(22,27)(23,30)(24,29)(31,37)(32,38)(33,36)(34,39)(35,40);(2,6)(5,7)(10,16)(15,27)(17,29)(18,30)(25,36)(26,39)(28,37)(38,40)

Polycyclic presentation:

2^9|c^2=gh,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,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, 7532050);