Finite Group Databasebeta

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#72321046

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#486293
commutator, Frattini, cosocleC2×M32C24
LCSG ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ #33432 ◁ G
LECSG ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

1249 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 1249 conjugacy classes of subgroups — too many to draw. Only its 151 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:

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

Polycyclic presentation:

2^10|b^2=fj,c^2=efghij,d^2=hj,e^2=hj,h^2=ij,i^2=j,b^a=bf,c^a=cfgj,c^b=ce,d^b=dhi,d^c=dg,e^a=eg,e^b=eghi,e^c=eg,e^d=ej,f^c=fgj,f^e=fj,g^b=gj,g^c=gj,h^b=hi,i^b=ij

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72321055, #72342920, #72342984
maximal normal subgroup of#72321055, #72342920, #72342984

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