#6574711

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
prime radical	C₂×#578	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1832 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)(9,10,14,12)(11,20,22,34)(13,18,30,26)(15,27,16,19)(17,24,25,23)(21,35,36,40)(28,32,29,31)(33,38,39,37);(1,3,4,8)(2,5,6,7)(9,10)(11,20)(12,14)(13,28)(15,19)(16,27)(17,24)(18,31)(21,33)(22,34)(23,25)(26,32)(29,30)(35,37)(36,39)(38,40);(9,11,14,22)(10,17,12,25)(13,21,30,36)(15,24,16,23)(18,33,26,39)(19,34,27,20)(28,35,29,40)(31,38,32,37);(9,12,14,10)(11,20,22,34)(13,29,30,28)(15,27,16,19)(17,23,25,24)(18,31,26,32)(21,33,36,39)(35,38,40,37);(9,13)(10,18)(11,21)(12,26)(14,30)(15,31)(16,32)(17,33)(19,29)(20,35)(22,36)(23,37)(24,38)(25,39)(27,28)(34,40)

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,d^b=di,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, 6481907);