#489999

order	512
nilpotency class	4
isoclinism	isoclinic class of #489745
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

734 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397195);