#490533

order	512
nilpotency class	4
isoclinism	isoclinic class of #490434
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#459
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

432 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 80 points:

(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,23,28)(19,24,30,45)(20,33,36,67)(21,37,41,66)(22,42,39,27)(25,48,50,34)(26,51,53,40)(29,55,56,78)(31,61,59,77)(32,64,60,47)(35,68,43,52)(38,54,69,49)(44,70,71,57)(46,74,73,63)(58,79,65,75)(62,76,80,72);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(17,19,23,30)(18,24,28,45)(20,29,36,56)(21,31,41,59)(22,32,39,60)(25,44,50,71)(26,46,53,73)(27,47,42,64)(33,55,67,78)(34,57,48,70)(35,58,43,65)(37,61,66,77)(38,62,69,80)(40,63,51,74)(49,72,54,76)(52,75,68,79);(17,20,23,36)(18,25,28,50)(19,29,30,56)(21,38,41,69)(22,43,39,35)(24,44,45,71)(26,52,53,68)(27,54,42,49)(31,62,59,80)(32,65,60,58)(33,66,67,37)(34,51,48,40)(46,75,73,79)(47,76,64,72)(55,77,78,61)(57,74,70,63);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(19,30)(24,45)(29,56)(31,59)(32,60)(44,71)(46,73)(47,64)(55,78)(57,70)(58,65)(61,77)(62,80)(63,74)(72,76)(75,79)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=i,d^2=g,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397729);