Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#1333 |
| commutator | C2×C16 | C22×C8 |
| Frattini, cosocle | #1311 | C23 |
| prime radical | C22×D16 | C8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ C2×#1311 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
398 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 105 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) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(2,7)(3,10)(9,15)(12,16);(1,2,4,9,5,10,13,16,6,7,11,15,14,3,8,12)(17,18,19,26,21,28,32,42,23,29,33,45,36,44,48,46,24,30,20,34,22,38,31,25,37,40,35,27,39,43,47,41);(1,3,11,16,5,2,8,15,6,10,4,12,14,7,13,9)(18,25,40,41,30,42,29,46)(19,31,20,32)(21,36)(22,39)(23,37)(26,43,45,38,34,44,27,28)(33,48,35,47)Polycyclic presentation:
2^10|b^2=d,c^2=degh,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,i^2=j,b^a=bhij,c^a=chij,c^b=ce,d^c=dg,e^b=eij,e^c=eg,e^d=ej,f^c=fi,g^b=gj,g^c=gi,h^c=hj,i^c=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72286351, #72287236, #72287247, … |
| maximal normal subgroup of | #72286351, #72287236, #72287247, … |
| maximal quotient of | #72287258, #72323154, #72323168, … |
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, 2, F.4);;
gap> SetPower(coll, 3, F.4*F.5*F.7*F.8);;
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, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);