#142974

order	512
nilpotency class	4
isoclinism	isoclinic class of #142972
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	#3493	C₂²
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

459 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 459 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 128

A minimal faithful permutation representation — 3 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=g,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fg,h^b=hi,d^c=dh,e^c=ei,f^c=fg,f^d=fi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50170);