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#72280088

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC4C2×#1655
commutator, Frattini, cosocle#376C24
socleC2C2×#7734
LCSG ▷ #376 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#376 ◁ G
LECSG ▷ #376 ▷ C22×C4 ▷ C2 ▷ C1

2941 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 2941 conjugacy classes of subgroups — too many to draw. Only its 219 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)(3,8)(4,9)(5,24)(6,29)(7,12)(10,28)(11,31)(13,32)(14,50)(15,44)(16,46)(17,36)(18,55)(19,45)(20,57)(21,42)(22,41)(23,43)(25,33)(26,39)(27,59)(30,35)(34,54)(37,51)(38,52)(40,58)(47,61)(48,60)(49,62)(53,64)(56,63);(1,3,7,17)(2,8,12,36)(4,18,21,51,23,56,22,53)(5,25)(6,30,31,16)(9,37,41,55,43,64,42,63)(10,44)(11,46,29,35)(13,19,47,26,49,20,48,58)(14,28)(15,52)(24,34)(27,54)(32,39,60,45,62,40,61,57)(33,59)(38,50);(1,4,5,26,7,23,27,58)(2,9,38,57)(3,13,14,51)(6,22,28,20,31,21,52,19)(8,32,34,63,36,62,33,55)(10,45,12,43)(11,42,59,39)(15,53,17,49)(16,48,25,56)(18,30,47,54)(24,40,29,41)(35,61,44,37,46,60,50,64);(1,5)(2,10,12,38)(3,14,17,15)(4,19,23,20)(6,28)(7,27)(8,33)(9,39)(11,24,29,59)(13,18)(16,25,30,54)(21,58,22,26)(31,52)(32,37,62,64)(34,36)(35,50)(40,43)(41,57)(42,45)(44,46)(47,53)(48,51)(49,56)(55,60,63,61)

Polycyclic presentation:

2^10|b^2=fhj,c^2=efhij,d^2=i,e^2=i,f^2=h,g^2=j,b^a=bf,c^a=cfg,c^b=ce,d^a=dg,d^b=dgij,d^c=dg,e^a=eg,e^b=egi,e^c=egj,f^a=fh,f^c=fg,f^d=fj,f^e=fj,g^a=gj,g^b=gj,h^c=hj,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72280134, #72280230
maximal normal subgroup of#72280134, #72280230

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