#7624637

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624184
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

1316 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)(11,23)(12,17)(13,22)(14,19)(15,30)(16,20)(18,27)(21,32)(24,34)(25,36)(26,39)(28,38)(29,33)(31,35)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,12)(10,17,19,23)(13,26,16,24)(15,25,29,27)(18,33,36,30)(20,39,22,34)(21,35,38,37)(28,31,32,40);(9,12,14,11)(10,18,19,36)(13,26,16,24)(15,31,29,40)(17,33,23,30)(20,37,22,35)(21,34,38,39)(25,32,27,28);(2,5)(6,8)(10,19)(17,23)(18,36)(20,22)(21,38)(30,33)(34,39)(35,37);(1,3,4,7)(2,6,5,8)(9,13,14,16)(10,20,19,22)(11,24,12,26)(15,28,29,32)(17,34,23,39)(18,37,36,35)(21,33,38,30)(25,40,27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531833);