#7611886

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

1961 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=h,e^2=f,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, 7519082);