#6584824

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574063
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
prime radical	#61224	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1771 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)(3,7)(4,8)(5,6)(9,10)(11,22)(12,16)(13,18)(14,19)(15,21)(17,24)(20,26)(23,39)(25,34)(27,32)(28,35)(29,36)(30,38)(31,37)(33,40);(2,6)(7,8)(9,11,14,12)(10,16,19,22)(13,28,29,40)(15,24,26,25)(17,21,34,20)(18,33,36,35)(23,31,27,30)(32,37,39,38);(1,3,5,4)(2,7,6,8)(9,11,15,24,14,12,26,25)(10,17,20,22,19,34,21,16)(13,23,30,40,29,27,31,28)(18,33,37,39,36,35,38,32);(1,4,5,3)(2,8,6,7)(9,12,14,11)(10,16,19,22)(13,28,29,40)(15,25,26,24)(17,21,34,20)(18,35,36,33)(23,31,27,30)(32,38,39,37);(9,13,14,29)(10,18,19,36)(11,23,12,27)(15,30,26,31)(16,32,22,39)(17,33,34,35)(20,37,21,38)(24,40,25,28)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,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=ehi,h^b=hi,d^c=dhi,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, 6492020);