#7611869

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563337
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

1915 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)(11,21)(12,26)(13,30)(14,31)(15,18)(16,20)(17,25)(19,28)(22,36)(23,34)(24,37)(27,40)(29,39)(32,35)(33,38);(1,3)(2,5)(4,7)(6,8)(9,11,14,25,15,22,16,32)(10,17,20,21,18,35,31,36)(12,27,13,24,28,38,29,23)(19,34,30,33,26,37,39,40);(2,6)(3,7)(10,18)(11,22)(12,28)(13,29)(20,31)(25,32)(33,40)(34,37);(2,6)(5,8)(9,12,16,29,15,28,14,13)(10,19,20,30,18,26,31,39)(11,23,25,27,22,24,32,38)(17,33,36,34,35,40,21,37);(9,13,14,28,15,29,16,12)(10,19,20,30,18,26,31,39)(11,24,25,38,22,23,32,27)(17,34,21,33,35,37,36,40)

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,d^c=di,e^c=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, 7519065);