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, socle | C2 | #135932 |
| commutator | C22×C16 | C22×C4 |
| Frattini, cosocle | C2×#346 | C23 |
| prime radical | C22×D16 | C2×C4 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ #11405 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
384 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 101 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:
(1,2,5,13,6,15,29,51,7,11,30,45,4,12,23,40,8,14,28,50,24,44,59,63,22,46,57,49,25,47,58,34)(3,16,18,9,19,10,26,38,20,43,27,39,31,41,61,48,21,54,56,37,32,42,60,64,33,52,62,35,17,53,55,36);(1,3)(2,9,11,35,14,37,46,39)(4,19)(5,26,28,60)(6,31)(7,33)(8,21)(10,40,41,51,42,34,53,63)(12,38,44,36,47,64,15,48)(13,43,49,54,50,52,45,16)(17,24)(18,29,56,59)(20,22)(23,27,58,62)(25,32)(30,55,57,61);(1,3)(2,10,11,41,14,42,46,53)(4,17)(5,27,28,62)(6,32)(7,20)(8,21)(9,34,35,63,37,40,39,51)(12,16,44,43,47,54,15,52)(13,48,49,38,50,36,45,64)(18,30,56,57)(19,24)(22,33)(23,26,58,60)(25,31)(29,55,59,61)Polycyclic presentation:
2^10|a^2=f,b^2=d,c^2=d,d^2=g,e^2=ij,f^2=h,h^2=i,i^2=j,b^a=bfhij,c^a=ce,c^b=cg,d^a=dij,e^b=ei,f^b=fh,f^c=fi,f^d=fj,g^a=gj,h^b=hi,h^c=hj,i^b=ij70 conjugacy classes — view the full character table.
| maximal subgroup of | #72225084, #72225085, #72225089, … |
| maximal normal subgroup of | #72225084, #72225085, #72225089, … |
| maximal quotient of | C2×#72225078 |
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, 3, F.4);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.9*F.10);;
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.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
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, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);