#489373

order	512
nilpotency class	4
isoclinism	isoclinic class of #489373
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

788 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:

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

Polycyclic presentation:

2^9|d^2=g,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gi,d^b=dg,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, 396569);