#7623370

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623311
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1657 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)(10,16)(11,21)(12,15)(14,27)(17,31)(19,34)(22,33)(23,39)(24,38)(25,26)(28,29)(30,36)(32,35)(37,40);(1,3,4,8)(2,5,6,7)(9,10,13,16)(11,22,24,17)(12,18,15,20)(14,19,26,30)(21,31,38,33)(23,37,29,32)(25,34,27,36)(28,40,39,35);(3,8)(5,7)(9,11)(10,17)(12,23)(13,24)(14,28)(15,29)(16,22)(18,32)(19,35)(20,37)(21,27)(25,38)(26,39)(30,40)(31,34)(33,36);(2,6)(5,7)(9,12,13,15)(10,18,16,20)(11,23,24,29)(14,25,26,27)(17,32,22,37)(19,34,30,36)(21,28,38,39)(31,40,33,35);(10,16)(17,22)(18,20)(19,30)(31,33)(32,37)(34,36)(35,40)

Polycyclic presentation:

2^9|b^2=f,d^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gh,h^a=hi,c^b=cf,e^b=ei,g^c=gh,h^c=hi

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, 7530566);