#6576951

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573635
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

2607 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)(11,17)(12,26)(13,19)(14,31)(15,20)(16,21)(18,29)(22,39)(23,34)(24,36)(25,37)(27,33)(28,35)(30,40)(32,38);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,27)(13,30)(14,24)(15,28)(16,25)(18,38)(19,40)(20,35)(21,37)(22,23)(26,33)(29,32)(31,36)(34,39);(9,12,16,30)(10,18,21,34)(11,22,25,32)(13,28,27,24)(14,23,15,29)(17,33,37,19)(20,40,31,26)(35,38,36,39);(1,4,5,3)(2,6,7,8)(9,11)(10,17)(12,13)(14,24)(15,28)(16,25)(18,39)(19,26)(20,35)(21,37)(22,29)(23,32)(27,30)(31,36)(33,40)(34,38);(9,13,16,27)(10,19,21,33)(11,23,25,29)(12,28,30,24)(14,22,15,32)(17,34,37,18)(20,38,31,39)(26,35,40,36)

Polycyclic presentation:

2^9|b^2=f,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=ehi,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, 6484147);