#6584326

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574070
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2004 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491522);