#6584316

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574084
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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2027 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)(3,8)(4,6)(5,7)(9,10)(11,21)(12,17)(13,18)(14,20)(15,25)(16,30)(19,28)(22,34)(23,37)(24,27)(26,33)(29,36)(31,35)(32,39)(38,40);(1,3,5,4)(2,6,7,8)(9,11,13,25)(10,15,18,21)(12,23,29,39)(14,26,28,27)(16,34,35,40)(17,32,36,37)(19,33,20,24)(22,30,38,31);(9,12,28,31,13,29,14,30)(10,16,20,36,18,35,19,17)(11,22,27,39,25,38,26,23)(15,32,24,34,21,37,33,40);(1,4,5,3)(2,6,7,8)(9,12,13,29)(10,17,18,36)(11,23,25,39)(14,30,28,31)(15,32,21,37)(16,19,35,20)(22,26,38,27)(24,34,33,40);(9,10)(11,24)(12,16)(13,18)(14,19)(15,26)(17,30)(20,28)(21,27)(22,34)(23,32)(25,33)(29,35)(31,36)(37,39)(38,40)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=f,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,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, 6491512);