#6585834

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574075
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

1890 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)(11,16)(12,18)(13,19)(14,21)(15,22)(17,26)(20,23)(24,32)(25,34)(27,35)(28,33)(29,38)(30,36)(31,37)(39,40);(2,6)(7,8)(9,11,15,12)(10,16,22,18)(13,28,31,39)(14,25,23,26)(17,21,34,20)(19,33,37,40)(24,30,27,29)(32,36,35,38);(1,3,5,4)(2,7,6,8)(9,12,14,26,15,11,23,25)(10,17,20,18,22,34,21,16)(13,27,29,28,31,24,30,39)(19,33,36,35,37,40,38,32);(1,3,5,4)(2,7,6,8)(9,12)(10,18)(11,15)(13,28)(14,26)(16,22)(17,21)(19,33)(20,34)(23,25)(24,29)(27,30)(31,39)(32,38)(35,36)(37,40);(1,4,5,3)(2,8,6,7)(9,13)(10,19)(11,24)(12,27)(14,29)(15,31)(16,32)(17,33)(18,35)(20,36)(21,38)(22,37)(23,30)(25,39)(26,28)(34,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,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=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, 6493030);