Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C162 · Schur multiplier (H2) = C32 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C162 | C23 |
| commutator, socle | C22 | C2×C162 |
| prime radical | C23 | C162 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C162 ◁ G | |
| LECS | G ▷ C162 ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 1024 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,3,8,4,9,11,20,5,10,12,21,14,22,23,30,6,7,13,17,15,18,24,27,16,19,25,28,26,29,31,32);(33,34,35,40,36,41,43,52,37,42,44,53,46,54,55,62,38,39,45,49,47,50,56,59,48,51,57,60,58,61,63,64);(2,7)(8,17)(9,18)(10,19)(20,27)(21,28)(22,29)(30,32)(34,39)(40,49)(41,50)(42,51)(52,59)(53,60)(54,61)(62,64)Polycyclic presentation:
2^11|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,i^2=k,c^a=cj,c^b=ck896 conjugacy classes — view the full character table.
| maximal subgroup of | #72225086, #72225108, #72225123, … |
| maximal normal subgroup of | #72225086, #72225108, #72225123, … |
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.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);