#489976

order	512
nilpotency class	4
isoclinism	isoclinic class of #489719
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

711 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:

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

Polycyclic presentation:

2^9|b^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,d^b=di,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397172);