Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #346518 |
| commutator, Frattini, cosocle | C2×#131 | C24 |
| LCS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
4021 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,16)(4,15)(5,11)(6,29)(7,13)(8,14)(9,20)(10,23)(12,32)(17,33)(18,47)(19,35)(21,42)(22,37)(24,56)(25,59)(26,40)(27,41)(28,43)(30,44)(31,45)(34,52)(36,55)(38,57)(39,58)(46,50)(48,53)(49,62)(51,61)(54,64)(60,63);(1,3,8,20)(2,9,14,16)(4,17,23,7)(5,24)(6,30,32,50)(10,33,15,13)(11,38)(12,44,29,46)(18,51)(19,31,53,22)(21,25)(26,52)(27,54)(28,57)(34,61)(35,37,48,45)(36,59)(39,42)(40,47)(41,63)(43,56)(49,60)(55,58)(62,64);(1,4,8,23)(2,10,14,15)(3,17,20,7)(5,25)(6,31,32,22)(9,33,16,13)(11,39)(12,45,29,37)(18,51)(19,30,53,50)(21,24)(26,52)(27,60)(28,58)(34,61)(35,46,48,44)(36,56)(38,42)(40,47)(41,64)(43,59)(49,54)(55,57)(62,63);(1,5,20,52,4,21,17,49,8,28,3,18,23,55,7,27)(2,11,9,34,15,42,33,62,14,43,16,47,10,36,13,41)(6,26,53,57,22,54,50,25,32,51,19,24,31,60,30,58)(12,40,35,38,45,64,44,59,29,61,48,56,37,63,46,39)Polycyclic presentation:
2^10|b^2=h,c^2=h,d^2=fi,e^2=i,f^2=ij,i^2=j,b^a=bj,c^a=cj,d^a=dg,d^b=de,d^c=defi,e^a=eh,e^b=ei,e^c=ej,f^b=fij,f^c=fj,g^b=ghj,g^c=ghj,g^e=gj,h^d=hj,i^b=ij67 conjugacy classes — view the full character table.
| maximal subgroup of | #72250237, #72250830 |
| maximal normal subgroup of | #72250237, #72250830 |
| maximal quotient of | #72250409, #72257291 |
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.8);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.6*F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6*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.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*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);