#151503

order	512
nilpotency class	5
isoclinism	isoclinic class of #151176
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	C₂²×C₁₆	C₂³
prime radical	#2215	C₂²
LCS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2151 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

195 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 195 conjugacy classes of subgroups — too many to draw. Only its 44 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:

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

Polycyclic presentation:

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

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, 58699);