#489392

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

787 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)(11,23)(12,27)(13,21)(14,20)(15,22)(16,24)(17,25)(18,29)(19,30)(26,33)(28,38)(31,37)(32,36)(34,40)(35,39);(3,7)(5,8)(9,11)(10,16)(12,28)(13,25)(14,32)(15,26)(17,18)(19,36)(20,34)(21,38)(22,35)(23,27)(24,30)(29,39)(31,40)(33,37);(1,2)(3,8)(4,6)(5,7)(9,11)(10,17)(12,24)(13,32)(14,25)(15,26)(16,18)(19,35)(20,33)(21,23)(22,36)(27,38)(28,30)(29,40)(31,39)(34,37);(9,12,13,29,15,31,14,30)(10,18,20,27,22,19,21,37)(11,24,32,40,26,39,25,28)(16,33,38,36,35,23,34,17)

Polycyclic presentation:

2^9|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, 396588);