Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #8541 |
| commutator | C26 | C22×C4 |
| Frattini, cosocle | C22×#120 | C23 |
| prime radical | #4930494 | C2 |
| LCS | G ▷ C26 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C22×#120 ▷ C24 ▷ C22 ▷ C1 | |
10135 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 10135 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(9,10)(11,14)(12,15)(13,16)(17,24)(18,22)(19,21)(20,23);(9,11)(10,14)(12,19)(13,18)(15,21)(16,22)(17,20)(23,24);(1,2,3,6,4,5,7,8)(9,12,10,15)(11,17,14,21)(13,20,16,23)(18,19,22,24)Polycyclic presentation:
2^10|c^2=d,d^2=i,c^a=cf,c^b=ce,d^a=dg,d^b=dgh,e^a=ej,e^c=egh,e^d=ej,f^b=fj,f^c=fg,f^d=fi,g^c=gi,h^c=hij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72233254, #72263076, #72263078, … |
| maximal normal subgroup of | #72233254, #72263076, #72263078, … |
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, 3, F.4);;
gap> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);