#7624999

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 ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1393 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:

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

Polycyclic presentation:

2^9|c^2=ghi,d^2=i,e^2=fi,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

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, 7532195);