#7624890

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

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

Polycyclic presentation:

2^9|b^2=i,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^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, 7532086);