#7624898

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 ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1326 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)(13,26)(14,27)(16,29)(17,32)(18,25)(20,28)(21,33)(22,34)(23,37)(24,30)(31,40)(35,36)(38,39);(1,3)(2,5)(4,8)(6,7)(9,10,15,12)(11,16,22,21)(13,17,28,24)(14,18,27,25)(19,29,34,33)(20,30,26,32)(23,31,35,38)(36,39,37,40);(9,11,20,37,14,23,13,19,15,22,26,36,27,35,28,34)(10,16,30,40,18,31,17,29,12,21,32,39,25,38,24,33);(3,8)(5,7)(9,12,15,10)(11,21,22,16)(13,24,28,17)(14,25,27,18)(19,33,34,29)(20,32,26,30)(23,38,35,31)(36,40,37,39);(2,6)(5,7)(10,12)(11,22)(17,24)(18,25)(19,34)(23,35)(30,32)(36,37)

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,e^d=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, 7532094);