#151602

order	512
nilpotency class	5
isoclinism	isoclinic class of #151520
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	#33922	C₂
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ #2169 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

185 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 185 conjugacy classes of subgroups — too many to draw. Only its 46 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)(17,18,22,27)(19,29,31,59)(20,34,36,58)(21,40,43,74)(23,44,42,28)(24,46,47,30)(25,50,52,38)(26,54,56,37)(32,63,62,79)(33,64,45,60)(35,55,67,80)(39,72,70,66)(41,51,75,78)(48,73,76,69)(49,61,57,68)(53,71,65,77);(1,3,6,14)(2,7,10,16)(4,13,5,12)(8,15,9,11)(17,19,22,31)(18,24,27,47)(20,35,36,67)(21,41,43,75)(23,45,42,33)(25,51,52,78)(26,55,56,80)(28,57,44,49)(29,58,59,34)(30,50,46,38)(32,54,62,37)(39,73,70,69)(40,48,74,76)(53,79,65,63)(60,66,64,72)(61,71,68,77);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(20,36)(25,52)(29,59)(30,46)(34,58)(35,67)(38,50)(39,70)(51,78)(53,65)(60,64)(61,68)(63,79)(66,72)(69,73)(71,77)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58798);