#131083

order	512
nilpotency class	4
isoclinism	isoclinic class of #130738
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

556 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 92 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 — 3 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=hi,e^2=i,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,e^c=ei,f^c=fg

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38279);