Finite Group Databasebeta

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#72343333

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#137479
commutatorC2×M32C22×C4
Frattini, cosocleC4×M32C23
prime radical#513022C2
LCSG ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ #11405 ◁ G
LECSG ▷ C4×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

409 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 77 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:

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

Polycyclic presentation:

2^10|a^2=d,d^2=g,e^2=hij,f^2=hij,h^2=i,i^2=j,b^a=bghij,c^a=cf,c^b=ce,d^b=dj,d^c=di,e^a=eghij,e^b=eh,e^c=eh,e^d=ej,f^a=fhij,f^b=fgh,f^c=fh,f^d=fj,f^e=fj,g^c=gj,h^a=hi,h^b=hij,h^c=hij,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72343339, #72343345, #72343437, …
maximal normal subgroup of#72343339, #72343345, #72343437, …

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