#6578661

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573643
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	#24280	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1886 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)(4,6)(5,8)(9,10)(11,17)(12,13)(14,16)(15,21)(18,30)(19,32)(20,34)(22,37)(23,28)(24,26)(25,38)(27,33)(29,31)(35,36)(39,40);(1,3,5,4)(2,6,8,7)(9,11,16,27)(10,17,14,33)(12,28,19,36)(13,23,32,35)(15,25,34,26)(18,37,29,39)(20,24,21,38)(22,31,40,30);(9,12)(10,18)(11,22)(13,21)(14,29)(15,31)(16,19)(17,35)(20,32)(23,33)(24,39)(25,28)(26,36)(27,40)(30,34)(37,38);(1,3,5,4)(2,7,8,6)(9,13,16,32)(10,19,14,12)(11,23,27,35)(15,29,34,18)(17,36,33,28)(20,31,21,30)(22,24,40,38)(25,39,26,37);(1,4,5,3)(2,6,8,7)(9,14)(10,16)(11,24)(12,29)(13,31)(15,21)(17,26)(18,19)(20,34)(22,39)(23,28)(25,33)(27,38)(30,32)(35,36)(37,40)

Polycyclic presentation:

2^9|b^2=fi,d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485857);