#6466304

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443050
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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1634 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)(9,10,15,18)(11,14,19,28)(12,22,25,16)(13,17,26,31)(20,33,30,24)(21,27,32,39)(23,35,37,29)(34,40,36,38);(9,11,14,18,15,19,28,10)(12,16,24,20,25,22,33,30)(13,17,27,21,26,31,39,32)(23,36,38,29,37,34,40,35);(1,3,4,8)(2,5,6,7)(9,12,14,24,15,25,28,33)(10,16,19,30,18,22,11,20)(13,23,27,38,26,37,39,40)(17,29,32,36,31,35,21,34);(9,13,14,27,15,26,28,39)(10,17,19,32,18,31,11,21)(12,23,24,38,25,37,33,40)(16,29,30,36,22,35,20,34);(13,26)(17,31)(21,32)(23,37)(27,39)(29,35)(34,36)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=ghi,d^2=h,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^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, 6373500);