#7611887

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563327
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1902 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)(3,5)(4,6)(7,8)(9,10)(11,24)(12,19)(13,20)(14,36)(15,38)(16,22)(17,23)(18,27)(21,35)(25,34)(26,32)(28,37)(29,39)(30,33)(31,40);(1,3)(2,5)(4,7)(6,8)(9,11,15,27,17,28,16,39)(10,18,22,24,23,29,38,37)(12,25,30,21,32,36,31,20)(13,33,14,19,35,40,34,26);(2,6)(3,7)(9,12)(10,19)(11,25)(13,29)(14,37)(15,30)(16,31)(17,32)(18,35)(20,39)(21,27)(22,40)(23,26)(24,34)(28,36)(33,38);(2,6)(5,8)(9,13,16,34,17,35,15,14)(10,20,22,25,23,21,38,36)(11,26,27,33,28,19,39,40)(12,29,31,24,32,18,30,37);(9,14,16,13,17,34,15,35)(10,21,38,25,23,20,22,36)(11,19,39,33,28,26,27,40)(12,24,31,18,32,37,30,29)

Polycyclic presentation:

2^9|b^2=f,d^2=h,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7519083);