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#72283371

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2C2×#13674
commutator, Frattini, cosocle#494C24
prime radical#512893C2
LCSG ▷ #494 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #3147 ◁ G
LECSG ▷ #494 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

1350 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1350 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=ghi,b^2=ghi,c^2=egij,d^2=h,e^2=h,f^2=j,g^2=i,h^2=j,c^a=ce,c^b=cf,d^b=dgij,d^c=dhj,e^a=ej,e^b=eg,e^c=eh,f^a=fgh,f^b=fh,f^c=fgj,f^d=fhj,f^e=fh,g^b=gi,g^c=gh,g^f=gi,h^b=hi,h^c=hj,h^f=hj,i^c=ij

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72283444, #72283908
maximal normal subgroup of#72283444, #72283908

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.7*F.8*F.9);;
gap> SetPower(coll, 2, F.7*F.8*F.9);;
gap> SetPower(coll, 3, F.5*F.7*F.9*F.10);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);