#6584809

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574077
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1780 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6492005);