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#72306809

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

1406 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1406 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)(4,18)(5,11)(6,12)(7,14)(8,13)(10,22)(15,37)(16,34)(17,36)(19,33)(20,52)(21,47)(23,53)(24,48)(25,60)(26,43)(27,44)(28,45)(29,46)(30,39)(31,35)(32,49)(38,56)(40,58)(41,51)(42,54)(50,63)(55,62)(57,61)(59,64);(1,3)(2,9)(4,19)(5,24,32,26,27,51,28,16)(6,29)(7,31)(8,17)(10,37)(11,41,49,34,44,48,45,43)(12,46)(13,36)(14,35)(15,22)(18,33)(20,55,50,54,56,59,57,25)(21,58)(23,30)(38,62,61,42,52,64,63,60)(39,53)(40,47);(1,4)(2,10)(3,15)(5,25,28,59,27,54,32,55)(6,30)(7,23)(8,22)(9,33)(11,42,45,62,44,60,49,64)(12,47)(13,18)(14,40)(16,50,51,20,26,57,24,56)(17,19)(21,31)(29,58)(34,61,41,38,43,63,48,52)(35,39)(36,37)(46,53);(1,5,7,32,8,27,29,28)(2,11,14,49,13,44,46,45)(3,16,6,24,17,26,31,51)(4,20,23,50,22,56,58,57)(9,34,12,48,36,43,35,41)(10,38,40,61,18,52,53,63)(15,25,30,55,19,54,21,59)(33,42,47,64,37,60,39,62)

Polycyclic presentation:

2^10|b^2=ghi,c^2=ghj,d^2=hij,e^2=hj,g^2=i,h^2=j,b^a=bij,c^a=cj,c^b=cj,d^b=df,d^c=de,e^b=eg,e^c=ej,e^d=eh,f^a=fj,f^b=fhj,f^c=fgh,f^d=fi,f^e=fh,g^b=gi,g^d=gh,g^f=gi,h^b=hi,h^d=hj,h^f=hj,i^d=ij

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72306820
maximal normal subgroup of#72306820

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