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)(3,14,19,44)(4,9)(5,10)(6,13)(7,11)(8,17,32,52)(12,27)(15,40)(16,47,50,43)(18,53,20,38)(21,54,55,61)(22,35)(23,42)(24,37)(25,33,45,41)(26,39)(28,36)(29,30)(31,56,60,48)(34,51,62,49)(46,64)(57,59)(58,63);(1,3,15,45,6,19,5,25)(2,8,30,14,13,32,39,44)(4,21,22,50,24,55,46,16)(7,18,26,52,27,20,29,17)(9,34,59,61,37,62,63,54)(10,38,11,33,40,53,12,41)(23,56,58,51,28,48,57,49)(31,36,47,64,60,42,43,35);(1,4)(2,9)(3,16)(5,22)(6,24)(7,28)(8,31)(10,35)(11,36)(12,42)(13,37)(14,43)(15,46)(17,48)(18,49)(19,50)(20,51)(21,45)(23,27)(25,55)(26,58)(29,57)(30,59)(32,60)(33,54)(34,38)(39,63)(40,64)(41,61)(44,47)(52,56)(53,62)Polycyclic presentation:
2^10|a^2=f,b^2=deij,d^2=j,e^2=ij,h^2=j,b^a=bd,c^a=cf,c^b=cdg,d^b=dh,e^a=ehj,e^b=eh,e^c=ei,e^d=eij,f^b=fj,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 | #72287010, #72287026, #72287086, … |
| maximal normal subgroup of | #72287010, #72287026, #72287086, … |
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, 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, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
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, 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);