#6575441

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573652
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

2071 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)(3,6)(4,5)(7,8)(9,10)(11,24)(12,29)(13,20)(14,21)(15,22)(16,34)(17,23)(18,27)(19,31)(25,37)(26,40)(28,36)(30,38)(32,39)(33,35);(2,5)(6,8)(9,11)(10,18)(12,30)(13,17)(14,31)(15,27)(16,28)(19,37)(20,34)(21,39)(22,24)(23,36)(25,32)(26,33)(29,35)(38,40);(9,12,15,26)(10,19,22,39)(11,25,27,14)(13,30,28,33)(16,32,17,31)(18,35,24,38)(20,21,36,37)(23,40,34,29);(1,3,4,7)(2,6,5,8)(9,13,15,28)(10,20,22,36)(11,17,27,16)(12,25,26,14)(18,34,24,23)(19,38,39,35)(21,29,37,40)(30,32,33,31);(9,14,15,25)(10,21,22,37)(11,26,27,12)(13,31,28,32)(16,33,17,30)(18,29,24,40)(19,36,39,20)(23,38,34,35)

Polycyclic presentation:

2^9|c^2=i,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=eh,h^b=hi,d^c=dh,g^c=gi,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, 6482637);