Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C82 · Schur multiplier (H2) = C16 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C82 | C43 |
| commutator | C42 | C4×C82 |
| Frattini, cosocle | C2×C162 | C23 |
| socle | C22 | #72216182 |
| prime radical | C23 | #122211 |
| LCS | G ▷ C42 ▷ C1 | |
| UCS | C1 ◁ C82 ◁ G | |
| LECS | G ▷ C2×C162 ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
1137 conjugacy classes of subgroups; only the 651 normal subgroups are shown.
This group has 1137 conjugacy classes of subgroups — too many to draw. Only its 651 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 1024 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,3,8,4,10,12,22,5,7,13,17,14,19,26,28,6,9,11,21,15,23,24,31,16,18,25,20,27,29,32,30);(33,34,35,40,36,42,44,54,37,39,45,49,46,51,58,60,38,41,43,53,47,55,56,63,48,50,57,52,59,61,64,62);(2,7,9,18)(3,11)(8,20,21,17)(10,19,23,29)(12,24)(13,25)(22,30,31,28)(26,32)(34,39,41,50)(35,43)(40,52,53,49)(42,51,55,61)(44,56)(45,57)(54,62,63,60)(58,64)Polycyclic presentation:
2^12|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,g^2=i,h^2=j,i^2=k,j^2=l,c^a=cik,c^b=cjl,d^c=dk,e^c=el,f^a=fk,f^b=fl640 conjugacy classes — view the full character table.
| maximal subgroup of | #72225124, #72225293, #72225307, … |
| maximal normal subgroup of | #72225124, #72225293, #72225307, … |
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", "l");;
gap> coll := SingleCollector(F, [2, 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, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.10*F.12);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);