Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#2403 |
| commutator, Frattini, cosocle | C23×C4 | C26 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
304244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 6 generators on 24 points:
(3,7)(5,8)(12,20)(16,21)(17,23)(18,24);(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,21)(13,14)(15,18)(16,20)(19,23)(22,24);(10,14)(12,20)(15,22)(18,24);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15,14,22)(12,18)(13,19)(16,17,21,23)(20,24);(2,6)(5,8)(10,14)(15,22)(16,21)(17,23);(3,7)(5,8)(10,14)(12,20)(17,23)(18,24)Polycyclic presentation:
2^11|d^2=j,g^2=k,b^a=bh,c^b=chk,d^a=di,d^b=dg,e^b=eik,f^b=fk,f^d=fij,g^a=gk,g^b=gk,g^c=gk,g^f=gk,h^d=hk,j^b=jk200 conjugacy classes — view the full character table.
| maximal subgroup of | #72237062, #72237527, #72238610, … |
| maximal normal subgroup of | #72237062, #72237527, #72238610, … |
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, 4, F.10);;
gap> SetPower(coll, 7, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);