#489978

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

712 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)(12,15)(13,18)(14,29)(16,26)(17,23)(19,32)(21,38)(22,35)(24,37)(25,40)(27,28)(30,34)(31,36)(33,39);(1,2)(3,6)(4,8)(5,7)(9,11,15,24)(10,17,12,21)(13,25,16,22)(14,30,31,39)(18,34,26,33)(19,35,27,40)(20,36,37,29)(23,28,38,32);(1,2)(3,6)(4,8)(5,7)(9,12,15,10)(11,21,24,17)(13,18,16,26)(14,27,31,19)(20,23,37,38)(22,33,25,34)(28,29,32,36)(30,40,39,35);(1,3,5,4)(2,6,7,8)(9,13,15,16)(10,18,12,26)(11,22,24,25)(14,28,31,32)(17,33,21,34)(19,29,27,36)(20,35,37,40)(23,39,38,30)

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,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, 397174);