#489390

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

784 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)(10,16)(12,25)(13,27)(14,29)(17,32)(18,30)(19,26)(20,36)(21,22)(23,39)(28,37)(31,34)(33,35)(38,40);(3,7)(5,8)(9,10)(11,17)(12,26)(13,18)(14,30)(15,19)(16,27)(20,29)(21,32)(22,38)(23,40)(24,34)(25,36)(28,33)(31,37)(35,39);(1,2)(3,8)(4,6)(5,7)(9,11)(10,17)(12,21)(13,28)(14,23)(15,24)(16,31)(18,33)(19,34)(20,35)(22,25)(26,32)(27,37)(29,39)(30,40)(36,38);(9,12,14,13,15,27,29,25)(10,18,20,26,19,36,30,16)(11,21,23,28,24,37,39,22)(17,33,35,32,34,38,40,31)

Polycyclic presentation:

2^9|d^2=g,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, 396586);