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#72279842

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC4#10078
commutator, Frattini, cosocleC2×#376C23
socleC2#102905
LCSG ▷ C2×#376 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#376 ◁ G
LECSG ▷ C2×#376 ▷ C22×C4 ▷ C2 ▷ C1

1746 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 1746 conjugacy classes of subgroups — too many to draw. Only its 141 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=j,b^2=fhi,c^2=efgj,d^2=h,e^2=i,f^2=hij,g^2=j,b^a=bf,c^a=cd,c^b=ce,d^a=dh,d^b=dg,d^c=dj,e^a=ei,e^b=egj,e^c=egij,e^d=ej,f^a=fhij,f^c=fgij,f^d=fj,f^e=fj,g^c=gj,h^b=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72279936, #72280134
maximal normal subgroup of#72279936, #72280134

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.10);;
gap> SetPower(coll, 2, F.6*F.8*F.9);;
gap> SetPower(coll, 3, F.5*F.6*F.7*F.10);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.8*F.9*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);