Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #12067 |
| commutator | C16 | C22×C16 |
| Frattini, cosocle | #1311 | C23 |
| prime radical | C2×D32 | C8 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ #7053 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
199 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,8,5,9,11,15,6,10,14,3,7,12,16,13)(18,23,25,34,26,35,38,37)(19,27,29,42)(20,30)(21,32)(24,33,39,36,40,46,48,47)(28,41,44,43)(31,45);(1,3,11,2,7,15,4,12,6,8,16,10,5,13,14,9)(18,24,38,48,26,40,25,39)(19,28,29,44)(20,31)(23,33,37,47,35,46,34,36)(27,41,42,43)(30,45);(17,18,19,23,20,24,27,36,21,25,28,37,30,39,43,33,22,26,29,35,31,40,42,47,32,38,44,34,45,48,41,46)Polycyclic presentation:
2^10|a^2=d,b^2=df,c^2=e,d^2=f,e^2=g,f^2=h,g^2=i,i^2=j,c^a=ce,c^b=cg,d^c=dij,e^a=eg,e^b=ei,e^d=ej,f^c=fj,g^a=gi,g^b=gj,i^a=ij94 conjugacy classes — view the full character table.
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.4);;
gap> SetPower(coll, 2, F.4*F.6);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 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, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);