#6574712

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

1857 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)(11,23,26,17)(12,19,24,33)(13,20,29,37)(15,18,16,22)(25,36,39,32)(27,35,28,40)(30,34,31,38);(1,3,4,8)(2,5,6,7)(9,11,14,26)(10,17,21,23)(12,16,24,15)(13,28,29,27)(18,33,22,19)(20,35,37,40)(25,31,39,30)(32,38,36,34);(10,18)(11,24)(12,26)(15,16)(19,33)(20,34)(21,22)(25,28)(27,39)(30,31)(35,40)(37,38);(9,12,14,24)(10,19,21,33)(11,15,26,16)(13,25,29,39)(17,22,23,18)(20,36,37,32)(27,31,28,30)(34,35,38,40);(9,13,14,29)(10,20,21,37)(11,25,26,39)(12,27,24,28)(15,30,16,31)(17,32,23,36)(18,34,22,38)(19,35,33,40)

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=i,e^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, 6481908);