#6467850

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443052
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1780 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)(3,6)(4,5)(7,8)(9,10,16,19)(11,14,22,15)(12,17,25,23)(13,18,31,28)(20,26,34,27)(21,29,35,30)(24,32,38,36)(33,39,37,40);(9,11,14,19,16,22,15,10)(12,23,26,34,25,17,27,20)(13,28,29,35,31,18,30,21)(24,37,39,32,38,33,40,36);(2,5)(6,8)(9,12,15,27,16,25,14,26)(10,17,11,20,19,23,22,34)(13,24,30,40,31,38,29,39)(18,32,21,33,28,36,35,37);(9,13,14,29,16,31,15,30)(10,18,22,35,19,28,11,21)(12,24,26,39,25,38,27,40)(17,32,34,37,23,36,20,33);(1,3,4,7)(2,6,5,8)(12,25)(17,23)(20,34)(24,38)(26,27)(32,36)(33,37)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375046);