#6466091

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442584
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
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1919 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=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, 6373287);