Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #9962 |
| commutator, Frattini, cosocle | C23×M16 | C23 |
| prime radical | #499916 | C2 |
| LCS | G ▷ C23×M16 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×M16 ▷ C22×C4 ▷ C2 ▷ C1 | |
1523 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 1523 conjugacy classes of subgroups — too many to draw. Only its 91 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(10,14,15,20)(11,16)(12,19)(17,24,23,21)(18,22);(1,2)(3,5,8,7)(4,6)(10,15)(17,23);(1,3)(2,5,6,7)(4,8)(9,10,16,21,12,17,18,20,13,15,22,24,19,23,11,14)Polycyclic presentation:
2^10|a^2=h,b^2=fh,c^2=dehi,e^2=i,i^2=j,b^a=bf,c^a=ce,c^b=cd,d^a=dg,d^b=dgh,d^c=dg,e^a=ei,e^b=eh,e^c=egj,e^d=ej,f^c=fghj,g^c=gj,h^c=hj,i^a=ij58 conjugacy classes — view the full character table.
| maximal subgroup of | #72307747 |
| maximal normal subgroup of | #72307747 |
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.8);;
gap> SetPower(coll, 2, F.6*F.8);;
gap> SetPower(coll, 3, F.4*F.5*F.8*F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);