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#72284385

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#233541
commutator, Frattini, cosocleC23×D4C24
prime radical#6437629C2
LCSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G
LECSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1

4705 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 4705 conjugacy classes of subgroups — too many to draw. Only its 207 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)(3,13,17,42)(4,9)(5,22)(6,27,30,11)(7,12)(8,16)(10,26)(14,43,18,40)(15,46)(19,35)(20,39,28,36)(21,37)(23,41)(24,54)(25,55,29,38)(31,45)(32,47,53,56)(33,49)(34,48)(44,62)(50,61)(51,59,52,60)(57,64,58,63);(1,3,7,17)(2,8)(4,18,21,14)(5,23)(6,16,30,49)(9,34)(10,32,22,53)(11,13)(12,33)(15,26)(19,50)(20,48,28,45)(24,44)(25,56)(27,42)(29,47)(31,37)(35,57,54,58)(36,40)(38,46,55,41)(39,43)(51,64)(52,63)(59,61,60,62);(1,4)(2,9)(3,14)(5,24)(6,28)(7,21)(8,31)(10,35)(11,39)(12,37)(13,40)(15,44)(16,48)(17,18)(19,26)(20,30)(22,54)(23,50)(25,51)(27,36)(29,52)(32,58)(33,34)(38,60)(41,62)(42,43)(45,49)(46,61)(47,64)(53,57)(55,59)(56,63);(1,5)(2,10,12,22)(3,15)(4,19,21,24)(6,29,30,25)(7,26)(8,32,33,53)(9,35)(11,38)(13,41)(14,44,18,50)(16,47,49,56)(17,23)(20,52)(27,55)(28,51)(31,57)(34,58)(36,59,39,60)(37,54)(40,61,43,62)(42,46)(45,63)(48,64)

Polycyclic presentation:

2^10|a^2=f,b^2=hj,d^2=ghi,e^2=j,b^a=be,c^a=ci,c^b=cj,d^a=dfg,d^b=dgh,d^c=dhi,e^a=eh,e^c=ej,e^d=ehi,f^b=fhj,f^d=fi,f^e=fj,g^a=gi,g^b=gj,g^c=gj,g^d=gj,g^e=gj,h^a=hj,h^d=hj,i^b=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72284400, #72284403, #72284775, …
maximal normal subgroup of#72284400, #72284403, #72284775, …

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