Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C216×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C26 | C25 |
| commutator | C25 | C24×C4 |
| prime radical | C24×D42 | C2 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C1 | |
843928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24);(2,4)(14,15);(6,7)(10,12)(14,15)(18,20)(21,22)(23,24);(18,20)(22,24);(1,2)(3,4)(5,7,8,6)(9,10)(11,12)(13,15,16,14)(17,18)(19,20)(21,22)(23,24)Polycyclic presentation:
2^11|a^2=f,e^2=fijk,b^a=bg,c^a=cj,d^a=di,d^c=dk,e^a=eh,e^b=eghijk,e^c=ej,e^d=ei320 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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 5, F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);