#6585819

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

1896 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)(9,10,14,19)(11,22,12,16)(13,18,30,37)(15,21,23,20)(17,26,35,25)(24,40,27,33)(28,34,29,36)(31,39,32,38);(2,6)(7,8)(9,11)(10,16)(12,14)(13,28)(15,25)(17,20)(18,36)(19,22)(21,35)(23,26)(24,32)(27,31)(29,30)(33,38)(34,37)(39,40);(1,3,5,4)(2,7,6,8)(9,11,23,26,14,12,15,25)(10,17,21,16,19,35,20,22)(13,24,32,29,30,27,31,28)(18,34,39,33,37,36,38,40);(1,3,5,4)(2,7,6,8)(9,12,14,11)(10,16,19,22)(13,29,30,28)(15,26,23,25)(17,21,35,20)(18,36,37,34)(24,32,27,31)(33,39,40,38);(1,4,5,3)(2,8,6,7)(9,13,14,30)(10,18,19,37)(11,24,12,27)(15,31,23,32)(16,33,22,40)(17,34,35,36)(20,38,21,39)(25,28,26,29)

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=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, 6493015);