#7624914

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624888
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 ◁ #525 ◁ #2929 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1310 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)(9,10)(11,14)(12,23)(13,18)(15,30)(16,21)(17,27)(19,31)(20,34)(22,24)(25,37)(26,36)(28,40)(29,32)(33,38)(35,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,10)(12,20,26,17)(13,18,16,21)(15,22,29,19)(23,27,36,34)(24,30,31,32)(25,33,28,35)(37,39,40,38);(9,12,24,38,16,28,15,27,14,26,31,39,13,25,29,34)(10,17,32,40,18,33,19,23,11,20,30,37,21,35,22,36);(3,8)(5,7)(9,13,14,16)(10,18,11,21)(12,25,26,28)(15,24,29,31)(17,33,20,35)(19,30,22,32)(23,37,36,40)(27,38,34,39);(2,6)(5,7)(10,11)(12,26)(18,21)(19,22)(25,28)(27,34)(30,32)(38,39)

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=i,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^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532110);