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#72341174

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C2×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#151391
commutator, Frattini, cosocle#1781C23
prime radicalC23×D4C2×D4
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

451 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 39 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)(3,13)(4,18)(5,10)(6,11,29,42)(7,12)(8,15)(9,22)(14,43,48,61)(16,50,27,44)(17,36)(19,54)(20,53)(21,35,59,40)(23,32)(24,34)(25,41,28,37)(26,38)(30,46,62,57)(31,49,39,51)(33,47)(45,64,58,60)(52,55)(56,63);(1,3,7,17)(2,8,12,32)(4,19,22,56)(5,23)(6,27)(9,33,18,52)(10,36)(11,39)(13,38)(14,45,48,58)(15,26)(16,29)(20,47)(21,57)(24,55)(25,51,28,49)(30,60,62,64)(31,42)(34,54)(35,43)(37,50,41,44)(40,61)(46,59)(53,63);(1,4)(2,9)(3,14,17,48)(5,24,26,20)(6,21,29,59)(7,22)(8,30,32,62)(10,34,38,53)(11,40,42,35)(12,18)(13,43)(15,46)(16,47)(19,49,56,51)(23,57)(25,45)(27,55)(28,58)(31,63)(33,44,52,50)(36,61)(37,60)(39,54)(41,64)

Polycyclic presentation:

2^10|a^2=f,b^2=fhj,c^2=fhi,d^2=f,e^2=j,b^a=bd,c^a=chij,c^b=cdej,d^a=df,d^b=dfi,d^c=dg,e^a=efghij,e^b=ehi,e^c=efghj,e^d=eh,f^c=fi,f^e=fj,g^a=gij,g^b=gh,g^c=gj,g^d=gi,g^e=gj,h^a=hi,h^c=hi,h^d=hj,i^b=ij

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341180
maximal normal subgroup of#72341180

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