#489393

order	512
nilpotency class	4
isoclinism	isoclinic class of #489390
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

755 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396589);