#6585829

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

1898 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=f,e^2=f,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=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, 6493025);