Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#2177 |
| commutator | C16 | C24×C4 |
| Frattini, cosocle | C2×C16 | C25 |
| prime radical | #7625195 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C4×#131 ◁ #33326 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
2496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,4,12,20,34,46,25,6,10,21,27,23,36,44,8,7,11,19,35,47,58,48,28,22,33,45,37,5,13,18,26)(3,9,15,30,42,54,62,49,17,29,43,51,39,55,59,24,14,31,38,57,60,64,63,52,40,53,61,56,16,32,41,50);(2,8)(4,18)(5,20)(6,22)(9,24)(10,25)(11,26)(12,36)(13,35)(15,41)(16,42)(17,40)(19,44)(21,48)(23,47)(27,34)(28,33)(29,49)(30,55)(31,50)(32,57)(37,58)(38,59)(39,60)(43,63)(45,46)(51,54)(52,53)(56,64)(61,62);(1,3)(2,9)(4,15)(5,16)(6,17)(7,14)(8,24)(10,29)(11,31)(12,30)(13,32)(18,41)(19,38)(20,42)(21,43)(22,40)(23,39)(25,49)(26,50)(27,51)(28,52)(33,53)(34,54)(35,57)(36,55)(37,56)(44,59)(45,61)(46,62)(47,60)(48,63)(58,64);(2,10,11,33)(4,19)(8,25,26,28)(9,29,31,53)(12,37,35,27)(13,34,36,58)(15,38)(18,44)(21,45)(24,49,50,52)(30,56,57,51)(32,54,55,64)(41,59)(43,61)(46,48)(62,63);(2,11)(3,14)(8,26)(10,33)(12,35)(13,36)(15,38)(16,39)(17,40)(25,28)(27,37)(34,58)(41,59)(42,60)(43,61)(62,63)Polycyclic presentation:
2^10|a^2=f,d^2=g,f^2=hi,h^2=ij,i^2=j,b^a=bfj,d^a=dij,e^a=ej,e^c=ej,f^b=fh,f^d=fj,g^a=gj,h^b=hi,i^b=ij130 conjugacy classes — view the full character table.
| maximal subgroup of | #72225018, #72225019, #72225087, … |
| maximal normal subgroup of | #72225018, #72225019, #72225087, … |
| maximal quotient of | #72232013, #72232026, #72234334, … |
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.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 6, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, 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);