Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×#120 |
| commutator | C24 | C22×C42 |
| Frattini, cosocle | C22×C42 | C24 |
| socle | C22 | C4×#386 |
| prime radical | C2×#525 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
3049 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,6,4,5,7,8)(10,14)(16,18)(17,21)(23,24);(2,5)(6,8)(9,10,15,21,13,14,12,17)(11,18,22,23,19,16,20,24);(9,11)(10,16)(12,20)(13,19)(14,18)(15,22)(17,23)(21,24);(16,18)(23,24)Polycyclic presentation:
2^10|a^2=e,b^2=ehj,e^2=i,h^2=j,b^a=bj,c^b=cij,d^b=df,d^c=dg,f^c=fij,g^b=gij184 conjugacy classes — view the full character table.
| maximal subgroup of | #72343216, #72343234, #72343259, … |
| maximal normal subgroup of | #72343216, #72343234, #72343259, … |
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.5);;
gap> SetPower(coll, 2, F.5*F.8*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);