#6514387

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

Polycyclic presentation:

2^9|b^2=fi,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, 6421583);