#489383

order	512
nilpotency class	4
isoclinism	isoclinic class of #489379
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

814 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)(11,16)(12,15)(13,28)(14,20)(17,34)(18,25)(19,26)(21,36)(22,29)(23,40)(24,33)(27,30)(31,39)(32,38)(35,37);(3,7)(5,8)(10,15)(12,25)(13,26)(16,29)(17,28)(18,19)(20,30)(21,37)(22,39)(23,38)(27,34)(31,32)(33,36)(35,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,22)(13,23)(14,24)(15,29)(17,35)(18,31)(19,32)(20,33)(21,27)(25,39)(26,38)(28,40)(30,36)(34,37);(9,12,26,34,14,27,13,25)(10,17,28,15,20,18,19,30)(11,21,38,39,24,22,23,37)(16,31,40,36,33,35,32,29)

Polycyclic presentation:

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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396579);