#6584340

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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2006 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)(22,33)(23,37)(24,26)(27,36)(29,35)(31,34)(32,40)(38,39);(1,3,5,4)(2,6,7,8)(9,11,13,25)(10,15,18,21)(12,23,29,40)(14,26,28,27)(16,33,34,38)(17,32,35,37)(19,24,20,36)(22,30,39,31);(9,12,28,31,13,29,14,30)(10,16,20,35,18,34,19,17)(11,22,27,40,25,39,26,23)(15,32,36,33,21,37,24,38);(1,4,5,3)(2,6,7,8)(9,12,13,29)(10,17,18,35)(11,23,25,40)(14,30,28,31)(15,32,21,37)(16,19,34,20)(22,26,39,27)(24,38,36,33);(9,10)(11,24)(12,16)(13,18)(14,19)(15,27)(17,30)(20,28)(21,26)(22,38)(23,37)(25,36)(29,34)(31,35)(32,40)(33,39)

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=ehi,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, 6491536);