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#72283274

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2C2×#13673
commutator, Frattini, cosocle#492C24
LCSG ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #3147 ◁ G
LECSG ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

1407 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1407 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)(3,9,18,36,20,37,46,42)(4,10)(5,26)(6,28)(7,14)(8,12)(11,32)(13,30)(15,31)(16,45,48,58,49,33,55,63)(17,19,35,52,51,41,27,34)(21,47,56,64,53,38,54,50)(22,57)(23,40)(24,61)(25,43)(29,59)(39,60)(44,62);(1,3,7,20)(2,9,15,36)(4,21)(5,27)(6,19,26,52)(8,18)(10,38,43,64)(11,41,30,34)(12,42,14,37)(13,35)(16,29,48,62)(17,32,51,28)(22,58,44,63)(23,53)(24,55,57,49)(25,56,60,54)(31,46)(33,59)(39,50,40,47)(45,61);(1,4)(2,10)(3,16,46,55,20,49,18,48)(5,22)(6,29)(7,23)(8,25)(9,33,42,58,37,45,36,63)(11,24)(12,43)(13,44)(14,40)(15,39)(17,50,27,38,51,64,35,47)(19,53,34,56,41,21,52,54)(26,57)(28,59)(30,62)(31,60)(32,61);(1,5,7,13)(2,11,14,6)(3,17,20,51)(4,22,23,44)(8,32,31,28)(9,34,37,52)(10,24,40,29)(12,30,15,26)(16,47,49,38)(18,27,46,35)(19,36,41,42)(21,33,53,45)(25,61,60,59)(39,57,43,62)(48,50,55,64)(54,63,56,58)

Polycyclic presentation:

2^10|a^2=gi,b^2=eg,c^2=gj,d^2=j,f^2=hj,g^2=ij,h^2=j,b^a=be,c^a=cij,c^b=cf,d^a=dhi,d^b=dhij,d^c=dgj,e^a=eh,e^b=ehj,e^c=eg,e^d=ej,f^a=fghij,f^b=fghij,f^c=fj,f^d=fhj,f^e=fhj,g^b=gh,g^d=gij,g^e=gij,h^a=hi,h^b=hi,h^d=hj,h^e=hj,i^b=ij

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72283316, #72284059
maximal normal subgroup of#72283316, #72284059

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