Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | C2×#117 |
| commutator | C2×C4 | C23×C16 |
| Frattini, cosocle | C2×C32 | C24 |
| socle | C2 | C2×#13640 |
| prime radical | C2×#41 | C2×C16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×#1397 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,4,11,5,13,23,43,6,14,24,44,26,47,66,91,7,10,25,36,27,38,67,80,29,39,68,81,69,84,109,117,8,12,22,42,28,45,63,89,30,46,64,90,70,92,106,121,31,37,65,41,71,82,107,87,72,83,108,88,110,118,126,120)(3,15,17,40,18,49,54,85,19,50,55,86,57,95,99,119,20,48,56,76,58,93,100,114,59,94,101,115,102,122,124,128,16,9,21,33,51,34,60,77,52,35,61,78,96,79,103,116,53,32,62,73,97,74,104,111,98,75,105,112,123,113,125,127);(1,3)(2,9)(4,21)(5,18)(6,19)(7,20)(8,16)(10,32)(11,40)(12,15)(13,34)(14,35)(17,22)(23,60)(24,61)(25,62)(26,57)(27,58)(28,51)(29,59)(30,52)(31,53)(33,42)(36,76)(37,48)(38,74)(39,75)(41,73)(43,85)(44,86)(45,49)(46,50)(47,79)(54,63)(55,64)(56,65)(66,103)(67,104)(68,105)(69,102)(70,96)(71,97)(72,98)(77,89)(78,90)(80,114)(81,115)(82,93)(83,94)(84,113)(87,111)(88,112)(91,119)(92,95)(99,106)(100,107)(101,108)(109,125)(110,123)(116,121)(117,128)(118,122)(120,127)(124,126);(3,16)(9,15)(17,21)(18,51)(19,52)(20,53)(32,48)(33,40)(34,49)(35,50)(54,60)(55,61)(56,62)(57,96)(58,97)(59,98)(73,76)(74,93)(75,94)(77,85)(78,86)(79,95)(99,103)(100,104)(101,105)(102,123)(111,114)(112,115)(113,122)(116,119)(124,125)(127,128);(2,10,12,37)(4,22)(9,32,15,48)(11,41,42,36)(13,38,45,82)(14,39,46,83)(17,21)(23,63)(24,64)(25,65)(33,76,40,73)(34,74,49,93)(35,75,50,94)(43,87,89,80)(44,88,90,81)(47,84,92,118)(54,60)(55,61)(56,62)(66,106)(67,107)(68,108)(77,114,85,111)(78,115,86,112)(79,113,95,122)(91,120,121,117)(99,103)(100,104)(101,105)(109,126)(116,128,119,127)(124,125)Polycyclic presentation:
2^10|a^2=e,d^2=f,e^2=g,g^2=h,h^2=i,i^2=j,b^a=bf,c^b=cj,d^a=dij,e^b=ej,e^d=ej,f^a=fj232 conjugacy classes — view the full character table.
| maximal subgroup of | #72321748, #72321842, #72321873, … |
| maximal normal subgroup of | #72321748, #72321842, #72321873, … |
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.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);