Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C217 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C25 | C26 |
| commutator | C24 | C25×C4 |
| prime radical | C23×#2403 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
At least 10790 conjugacy classes of subgroups; only the 12720 normal subgroups are shown.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(14,16)(18,21)(19,23);(5,6)(7,8)(9,10,12,11)(13,14)(15,16)(17,18)(19,22)(20,21)(23,24);(18,21)(22,24);(2,4)(6,8)(10,11);(5,6)(7,8)(9,11,12,10)(17,18)(19,22)(20,21)(23,24);(1,2)(3,4)(5,6)(7,8)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^11|b^2=g,e^2=g,b^a=bj,c^b=cjk,d^b=dh,e^a=ejk,e^c=ejk,e^d=eh,f^a=fjk,f^d=fhi332 conjugacy classes — view the full character table.
| maximal subgroup of | #72330739 |
| maximal normal subgroup of | #72330739 |
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, 2, F.7);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.10*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.10*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);