#6464099

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	C₂×#535	C₂²
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) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=g,d^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,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, 6371295);