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#72319074

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#129029
commutatorC2×C16C2×C42
Frattini, cosocleC4×M32C23
prime radical#33748C22
LCSG ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ #11406 ◁ G
LECSG ▷ C4×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

356 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 97 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:

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

Polycyclic presentation:

2^10|b^2=d,c^2=efij,d^2=g,e^2=hi,f^2=i,g^2=j,h^2=ij,i^2=j,b^a=bghi,c^a=ce,c^b=cghj,d^a=di,d^c=di,e^a=eh,e^b=ei,e^c=egi,e^d=ej,f^a=fg,f^b=fj,f^c=fgi,f^e=fj,g^a=gj,g^c=gj,h^a=hi,h^b=hj,i^a=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72319087, #72343434, #72343439
maximal normal subgroup of#72319087, #72343434, #72343439

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