#489954

order	512
nilpotency class	4
isoclinism	isoclinic class of #489711
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

738 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)(9,10)(11,22)(12,18)(13,19)(14,28)(15,32)(16,21)(17,24)(20,30)(23,37)(25,36)(26,35)(27,38)(29,33)(31,39)(34,40);(1,2)(3,6)(4,8)(5,7)(9,11,16,12)(10,17,21,35)(13,23,15,25)(14,29,31,40)(18,28,22,39)(19,33,32,34)(20,36,38,37)(24,27,26,30);(1,2)(3,6)(4,8)(5,7)(9,12,16,11)(10,18,21,22)(13,23,15,25)(14,26,31,24)(17,28,35,39)(19,37,32,36)(20,34,38,33)(27,29,30,40);(1,3,5,4)(2,6,7,8)(9,13,16,15)(10,19,21,32)(11,23,12,25)(14,27,31,30)(17,33,35,34)(18,36,22,37)(20,28,38,39)(24,29,26,40)

Polycyclic presentation:

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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397150);