#6520044

order	512
nilpotency class	3
isoclinism	isoclinic class of #6502937
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

1700 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,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, 6427240);