Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #490518 |
| commutator | C4×C16 | C23×C4 |
| Frattini, cosocle | C2×C4×C16 | C24 |
| socle | C2 | #72216276 |
| prime radical | #72225339 | C2 |
| LCS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C42×C8 ◁ C8×#346 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13)(3,15,18,45)(4,23,24,58)(5,12,28,35)(6,14,27,10)(8,29,32,19)(9,33,34,25)(11,37,40,16)(17,50,51,63)(20,44,52,64)(21,46,22,43)(26,57,48,39)(30,55,42,54)(31,60,61,53)(36,49,38,56)(41,59,62,47);(1,3,16,47,5,17,48,54,27,21,56,60,33,52,24,29,7,18,37,59,28,51,26,55,6,22,49,53,25,20,4,19)(2,8,23,44,9,31,38,43,14,42,57,63,35,62,40,45,13,32,58,64,34,61,36,46,10,30,39,50,12,41,11,15);(2,9,14,35,13,34,10,12)(3,17,21,52,18,51,22,20)(11,23,38,57,40,58,36,39)(19,47,54,60,29,59,55,53);(2,10,13,14)(3,18)(4,24)(8,30,32,42)(9,12,34,35)(11,38,40,36)(15,43,45,46)(16,37)(17,51)(20,52)(21,22)(23,57,58,39)(26,48)(31,41,61,62)(44,63,64,50)(49,56)Polycyclic presentation:
2^11|a^2=k,b^2=eh,c^2=fhi,d^2=g,e^2=hj,f^2=ik,h^2=jk,i^2=k,j^2=k,b^a=be,c^a=chj,c^b=cf,d^a=djk,d^b=dijk,e^a=eh,e^d=ek,f^b=fi,g^a=gk,h^a=hj,i^b=ik,j^a=jk164 conjugacy classes — view the full character table.
| maximal subgroup of | #72225374, #72226910, #72226936, … |
| maximal normal subgroup of | #72225374, #72226910, #72226936, … |
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.11);;
gap> SetPower(coll, 2, F.5*F.8);;
gap> SetPower(coll, 3, F.6*F.8*F.9);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 6, F.9*F.11);;
gap> SetPower(coll, 8, F.10*F.11);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);