#6584821

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574063
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

1801 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)(13,19)(14,20)(15,30)(16,21)(18,24)(23,39)(25,32)(26,33)(27,34)(28,36)(29,38)(31,37)(35,40);(2,6)(7,8)(9,11,14,12)(10,17,20,22)(13,27,28,40)(15,24,16,26)(18,30,33,21)(19,35,36,34)(23,29,25,31)(32,38,39,37);(1,3,5,4)(2,7,6,8)(9,11,16,26,14,12,15,24)(10,18,30,17,20,33,21,22)(13,23,31,40,28,25,29,27)(19,34,38,32,36,35,37,39);(1,4,5,3)(2,8,6,7)(9,12,14,11)(10,17,20,22)(13,27,28,40)(15,26,16,24)(18,30,33,21)(19,34,36,35)(23,29,25,31)(32,37,39,38);(9,13,14,28)(10,19,20,36)(11,23,12,25)(15,29,16,31)(17,32,22,39)(18,34,33,35)(21,37,30,38)(24,27,26,40)

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=dhi,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, 6492017);