#6583354

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
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1638 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,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, 6490550);