#6491459

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470028
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2929 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:

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

Polycyclic presentation:

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

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, 6398655);