#489356

order	512
nilpotency class	4
isoclinism	isoclinic class of #489355
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 ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

783 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:

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

Polycyclic presentation:

2^9|c^2=i,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,e^c=eh,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396552);