#6491461

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

2975 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^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, 6398657);