#489395

order	512
nilpotency class	4
isoclinism	isoclinic class of #489394
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 ◁ #2965 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

768 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)(12,18)(13,28)(14,31)(15,20)(16,23)(17,27)(19,30)(22,35)(24,36)(25,40)(26,37)(29,38)(32,33)(34,39);(3,7)(5,8)(9,11,15,24)(10,16,20,34)(12,23,26,39)(13,29,27,22)(14,32,30,25)(17,36,28,21)(18,33,37,40)(19,38,31,35);(1,2)(3,8)(4,6)(5,7)(12,26)(13,27)(17,28)(18,37)(21,36)(22,29)(23,39)(33,40);(9,12,14,27,15,26,30,13)(10,17,19,18,20,28,31,37)(11,22,25,23,24,29,32,39)(16,33,35,36,34,40,38,21)

Polycyclic presentation:

2^9|b^2=i,d^2=g,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,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, 396591);