#489003

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
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 ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

734 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:

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

Polycyclic presentation:

2^9|b^2=hi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,g^a=gi,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, 396199);