Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #8266 |
| commutator | C4×C8 | C2×C4×C8 |
| Frattini, cosocle | C4×#346 | C23 |
| socle | C2 | #72225535 |
| prime radical | C23 | #12667 |
| LCS | G ▷ C4×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×#1398 ◁ C4×#1398 ◁ G | |
| LECS | G ▷ C4×#346 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,4,13,7,16,6,12,10,18,28,15,9,17,8,37)(3,19,21,40,24,45,26,43,25,41,38,42,36,51,23,11)(5,29,31,44,20,14,33,48,35,49,27,59,34,50,32,46)(22,54,55,39,30,61,56,47,57,64,53,60,62,52,58,63);(1,3,20,22,8,38,27,55,9,24,5,30,28,23,33,56,10,25,34,57,6,21,31,53,7,36,35,62,4,26,32,58)(2,11,29,60,12,19,46,61,17,40,50,63,13,45,59,64,18,43,49,39,37,41,48,52,16,42,14,47,15,51,44,54);(2,12,17,13,18,37,16,15)(3,21,24,26,25,38,36,23)(14,44,29,46,50,59,49,48)(22,53,30,58,57,55,62,56)Polycyclic presentation:
2^11|a^2=d,b^2=dehjk,c^2=fhj,d^2=g,e^2=h,f^2=ik,g^2=k,h^2=j,i^2=k,j^2=k,b^a=bhk,c^a=cf,c^b=cf,d^b=dj,e^a=ej,e^b=ej,e^d=ek,f^a=fi,f^b=fi,g^b=gk,h^a=hk,h^b=hk,i^a=ik,i^b=ik224 conjugacy classes — view the full character table.
| maximal subgroup of | #72225554, #72226993, #72227105, … |
| maximal normal subgroup of | #72225554, #72226993, #72227105, … |
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.4);;
gap> SetPower(coll, 2, F.4*F.5*F.8*F.10*F.11);;
gap> SetPower(coll, 3, F.6*F.8*F.10);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9*F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);