Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C27 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
At least 6300 conjugacy classes of subgroups; only the 38952 normal subgroups are shown.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 24 points:
(5,6)(7,8)(17,18)(19,23)(20,22)(21,24);(1,2)(3,4)(9,10)(11,15)(12,14)(13,16);(2,4)(6,8);(9,11)(10,13)(12,16)(14,15)(17,19)(18,21)(20,24)(22,23);(10,14)(13,15)(18,22)(21,23);(11,16)(13,15);(19,24)(21,23)Polycyclic presentation:
2^11|c^a=ci,c^b=ch,d^a=dk,d^b=dj,e^a=ek,e^b=ej,f^d=fj,g^d=gk284 conjugacy classes — view the full character table.
| maximal subgroup of | #72225731, #72225838, #72226177, … |
| maximal normal subgroup of | #72225731, #72225838, #72226177, … |
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> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);