#489374

order	512
nilpotency class	4
isoclinism	isoclinic class of #489373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,d^2=g,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396570);