#6574585

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	C₂×#577	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1851 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)(9,10,14,21)(11,23,25,17)(12,27,26,19)(13,20,30,37)(15,18,16,22)(24,36,29,33)(28,40,39,35)(31,34,32,38);(1,3,4,8)(2,5,6,7)(9,11,14,25)(10,17,21,23)(12,15,26,16)(13,28,30,39)(18,27,22,19)(20,35,37,40)(24,32,29,31)(33,38,36,34);(10,18)(11,12)(15,16)(19,27)(20,34)(21,22)(24,28)(25,26)(29,39)(31,32)(35,40)(37,38);(9,12)(10,19)(11,15)(13,29)(14,26)(16,25)(17,18)(20,36)(21,27)(22,23)(24,30)(28,31)(32,39)(33,37)(34,35)(38,40);(9,13)(10,20)(11,24)(12,28)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(21,37)(22,38)(23,36)(25,29)(26,39)(27,40)

Polycyclic presentation:

2^9|a^2=f,b^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, 6481781);