Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #365433 |
| commutator | C22×C8 | C23×C4 |
| Frattini, cosocle | C23×C8 | C24 |
| prime radical | C2×#3414 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ #10711 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
2152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(3,13,16,44)(5,23)(6,28)(8,31,33,14)(9,18)(10,24)(12,29)(15,43,49,50)(17,25,30,45)(19,55,51,61)(20,57)(22,48)(26,59)(27,38)(32,46,63,47)(34,39,42,52)(35,64,53,56)(36,58)(40,62)(41,60);(1,2,7,11)(3,14)(4,18,21,9)(5,24,27,60)(6,29,28,12)(8,13)(10,38,41,23)(15,47)(16,31)(17,52)(19,56)(20,58,26,62)(22,37,48,54)(25,34)(30,39)(32,50)(33,44)(35,61)(36,57,40,59)(42,45)(43,63)(46,49)(51,64)(53,55);(1,3)(2,8)(4,19)(5,25)(6,30)(7,16)(9,35)(10,39)(11,33)(12,42)(13,38)(14,24)(15,48)(17,28)(18,53)(20,43)(21,51)(22,49)(23,44)(26,50)(27,45)(29,34)(31,60)(32,54)(36,47)(37,63)(40,46)(41,52)(55,59)(56,58)(57,61)(62,64);(1,4,7,21)(2,9,11,18)(3,15)(5,26,27,20)(6,22,28,48)(8,32)(10,40,41,36)(12,37,29,54)(13,43)(14,46)(16,49)(17,51)(19,30)(23,59,38,57)(24,58,60,62)(25,61)(31,47)(33,63)(34,53)(35,42)(39,64)(44,50)(45,55)(52,56)Polycyclic presentation:
2^10|a^2=e,b^2=ej,d^2=ej,f^2=i,i^2=j,b^a=bg,c^a=cf,c^b=cfj,d^a=dh,d^b=dj,d^c=dij,e^c=ej,f^a=fi,f^b=fi,f^c=fij,f^d=fj,g^d=gj,h^b=hj,i^a=ij,i^b=ij,i^c=ij94 conjugacy classes — view the full character table.
| maximal subgroup of | #72320350, #72320733, #72343936 |
| maximal normal subgroup of | #72320350, #72320733, #72343936 |
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, 2, F.5*F.10);;
gap> SetPower(coll, 4, F.5*F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);