#6491489

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470237
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	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3082 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)(11,15)(12,22,25,16)(13,26)(14,18)(17,28)(19,29,23,32)(20,34)(21,31)(24,33,38,36)(27,30)(35,40,37,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,23)(13,27)(14,21)(16,32)(17,34)(18,31)(19,25)(20,28)(22,29)(24,35)(26,30)(33,39)(36,40)(37,38);(2,5)(6,8)(9,12)(10,16)(11,19)(13,24)(14,25)(15,29)(17,33)(18,22)(20,35)(21,23)(26,36)(27,37)(28,38)(30,39)(31,32)(34,40);(2,5)(6,8)(9,13,14,28)(10,17,18,26)(11,20,21,27)(12,24,25,38)(15,30,31,34)(16,33,22,36)(19,35,23,37)(29,39,32,40);(1,3,4,7)(2,6,5,8)(10,18)(15,31)(16,22)(17,26)(29,32)(30,34)(33,36)(39,40)

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=dh,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, 6398685);