#6574584

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573652
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	C₂×#578	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1841 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^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=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, 6481780);