Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #147555 |
| commutator | C22×#41 | C22×C4 |
| Frattini, cosocle | C2×#532 | C23 |
| prime radical | #6437453 | C2 |
| LCS | G ▷ C22×#41 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ C2×#3500 ◁ G | |
| LECS | G ▷ C2×#532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
3608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,13)(4,9)(5,24)(6,12)(7,28)(8,16)(10,14)(11,27)(15,48)(17,45)(18,44)(19,41)(20,54)(21,57)(22,38)(23,39)(25,34)(26,42)(29,40)(30,31)(32,56)(33,53)(35,46)(36,52)(37,47)(43,51)(49,61)(50,63)(55,62)(58,60)(59,64);(1,3,14,46,6,18,5,25)(2,8,31,44,12,33,42,13)(4,20,21,51,23,55,47,15)(7,17,26,53,27,19,30,16)(9,36,60,54,39,63,64,62)(10,41,11,34,24,45,28,35)(22,56,59,52,29,49,58,50)(32,40,43,37,61,38,48,57);(1,4)(2,9)(3,15)(5,21)(6,23)(7,29)(8,32)(10,37)(11,38)(12,39)(13,43)(14,47)(16,49)(17,50)(18,51)(19,52)(20,46)(22,27)(24,57)(25,55)(26,59)(28,40)(30,58)(31,60)(33,61)(34,54)(35,62)(36,45)(41,63)(42,64)(44,48)(53,56)Polycyclic presentation:
2^10|b^2=deij,d^2=j,e^2=ij,h^2=j,b^a=bd,c^a=cf,c^b=cdg,d^a=dj,d^b=dh,e^a=eh,e^b=eh,e^c=ei,e^d=eij,f^b=fj,g^a=gj,g^b=ghi,g^d=gj,g^e=gij,h^a=hj,h^b=hi,h^c=hj,h^d=hj,h^g=hj,i^a=ij,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72286971, #72287007, #72287068, … |
| maximal normal subgroup of | #72286971, #72287007, #72287068, … |
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*F.5*F.9*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);