#7611885

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

1905 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)(11,14)(12,18)(13,19)(15,21)(16,22)(17,24)(20,32)(23,31)(25,30)(26,34)(27,39)(28,37)(29,38)(33,35)(36,40);(1,3)(2,5)(4,7)(6,8)(9,11,15,24,16,25,32,19)(10,17,20,14,22,13,21,30)(12,23,28,35,29,39,40,34)(18,33,36,31,38,26,37,27);(2,6)(3,7)(9,12)(10,18)(11,23)(13,26)(14,31)(15,28)(16,29)(17,33)(19,34)(20,36)(21,37)(22,38)(24,35)(25,39)(27,30)(32,40);(2,6)(5,8)(9,13,15,30,16,17,32,14)(10,19,21,25,22,24,20,11)(12,26,28,27,29,33,40,31)(18,34,37,39,38,35,36,23);(9,14,15,13,16,30,32,17)(10,19,20,11,22,24,21,25)(12,27,28,33,29,31,40,26)(18,35,36,39,38,34,37,23)

Polycyclic presentation:

2^9|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, 7519081);