Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #12020 |
| commutator | C22×C16 | C22×C4 |
| Frattini, cosocle | C2×#346 | C23 |
| socle | C2 | #135846 |
| prime radical | C24 | #365 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C4×C8 ◁ C22×#346 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
392 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,4,10,5,12,20,38,6,9,21,32,23,34,49,60,7,11,19,37,24,39,47,63,26,33,48,36,53,61,64,62)(3,13,15,8,16,29,18,30,17,35,44,31,22,51,46,40,14,41,42,27,43,56,45,57,25,54,55,28,52,58,50,59);(1,3)(2,8,9,28,11,27,33,31)(4,18,19,45)(5,22)(6,25)(7,14)(10,35,36,41,37,54,32,13)(12,40,34,30,39,59,61,57)(15,20,42,47)(16,23)(17,26)(21,50,48,46)(24,52)(29,60,51,38,56,62,58,63)(43,53)(44,64,55,49);(8,27)(13,41)(28,31)(29,56)(30,57)(35,54)(40,59)(51,58)Polycyclic presentation:
2^10|a^2=f,b^2=d,d^2=g,f^2=h,h^2=i,i^2=j,b^a=bfhij,c^a=ce,c^b=cg,d^a=dij,e^b=ej,f^b=fh,f^d=fj,g^a=gj,h^b=hi,i^b=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72224982, #72225074, #72225085, … |
| maximal normal subgroup of | #72224982, #72225074, #72225085, … |
| maximal quotient of | C2×#72224972 |
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);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, 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);