Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #1368 |
| commutator | C16 | C82 |
| Frattini, cosocle | C4×#346 | C22 |
| socle | C2 | #93269 |
| prime radical | C23 | #1385 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C42×C8 ◁ C4×#346 ◁ G | |
| LECS | G ▷ C4×#346 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
290 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 135 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 — 2 generators on 64 points:
(1,2,4,11,6,14,25,44,8,13,26,15,9,36,7,34)(3,16,17,49,20,51,23,59,19,52,35,45,22,50,21,58)(5,27,29,47,30,62,31,46,32,43,24,60,33,64,61,12)(10,37,38,18,40,54,48,56,41,28,63,55,39,53,42,57);(1,3,7,35,9,20,26,21,8,19,25,17,6,22,4,23)(2,10,15,48,14,39,34,38,13,41,11,42,36,40,44,63)(5,28,61,18,33,53,24,56,32,37,31,55,30,54,29,57)(12,45,43,16,47,59,64,50,46,49,27,52,60,58,62,51)Polycyclic presentation:
2^10|a^2=c,b^2=ceh,c^2=f,d^2=g,e^2=h,f^2=j,g^2=i,i^2=j,b^a=bd,c^b=cij,d^a=dg,d^b=dg,d^c=dj,e^a=eij,e^b=eij,f^b=fj,g^a=gi,g^b=gi,h^a=hj,h^b=hj,i^a=ij,i^b=ij136 conjugacy classes — view the full character table.
| maximal subgroup of | #72225326, #72225330, #72227096, … |
| maximal normal subgroup of | #72225326, #72225330, #72227096, … |
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.3);;
gap> SetPower(coll, 2, F.3*F.5*F.8);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*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> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);