#6514391

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500489
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

1717 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=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, 6421587);