Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #72240171 |
| commutator, Frattini, cosocle | C2×#535 | C24 |
| LCS | G ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35527 ◁ G | |
| LECS | G ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
5779 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,21)(8,23)(9,18)(13,29)(14,31)(22,27)(24,30)(26,28);(2,7)(3,12)(5,21)(8,24)(9,18)(13,22)(14,31)(15,25)(16,32)(17,19)(23,30)(27,29);(1,2,6,7)(3,13,16,30)(4,18)(5,11)(8,25,27,19)(9,20)(10,21)(12,22,32,23)(14,28,31,26)(15,29,17,24);(1,3,14,32,6,17,31,25)(2,8,26,30,7,22,28,29)(4,19,5,16,20,12,21,15)(9,24,11,27,18,13,10,23)Polycyclic presentation:
2^11|c^2=gk,d^2=ehijk,i^2=k,c^a=cg,c^b=chj,d^a=df,d^b=de,d^c=defijk,e^a=eh,e^c=eh,e^d=ek,f^b=fh,f^c=fi,f^d=fhj,g^d=ghij,g^e=gk,g^f=gjk,h^c=hk,i^a=ij,i^b=ik,i^c=ij,i^f=ik,i^g=ik,j^c=jk,j^d=jk62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319061, #72338531, #72338574, … |
| maximal normal subgroup of | #72319061, #72338531, #72338574, … |
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, 3, F.7*F.11);;
gap> SetPower(coll, 4, F.5*F.8*F.9*F.10*F.11);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.11);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> SetConjugate(coll, 10, 4, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);