#151600

order	512
nilpotency class	5
isoclinism	isoclinic class of #151512
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	C₂×D₄	D₁₆
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

185 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 185 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,d^2=fh,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,h^a=hi,d^b=df,e^b=eg,f^b=fh,h^b=hi,d^c=dg,e^c=eg

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, 58796);