Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #140751 |
| commutator | C4×C16 | C22×C8 |
| Frattini, cosocle | C2×#1311 | C23 |
| prime radical | #513082 | C4 |
| LCS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ C2×#7053 ◁ G | |
| LECS | G ▷ C2×#1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
561 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 561 conjugacy classes of subgroups — too many to draw. Only its 132 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:
(1,2,4,9,6,11,19,27,7,12,20,28,5,10,18,26)(3,13,14,30,15,24,31,25,16,29,17,32,21,8,22,23)(34,39,42,44,43,54,45,55)(35,46,47,49)(36,50)(40,52,56,53,57,63,61,64)(41,58,59,60)(48,62);(1,3)(2,8)(4,17)(5,21)(6,15)(7,16)(9,25)(10,29)(11,13)(12,24)(14,20)(18,31)(19,22)(23,28)(26,30)(27,32)(34,40)(35,41)(36,48)(37,51)(39,52)(42,56)(43,57)(44,53)(45,61)(46,58)(47,59)(49,60)(50,62)(54,63)(55,64);(33,34,41,52,36,44,46,61,51,42,60,64,62,54,47,57,38,43,59,63,50,55,49,56,37,45,58,53,48,39,35,40)Polycyclic presentation:
2^11|a^2=d,c^2=eh,d^2=g,e^2=hi,f^2=j,g^2=j,h^2=ik,i^2=k,b^a=bf,c^a=chi,c^b=ce,d^b=dj,d^c=di,e^a=eik,e^b=eh,e^d=ek,f^b=fj,g^c=gk,h^a=hk,h^b=hi,i^b=ik116 conjugacy classes — view the full character table.
| maximal subgroup of | #72225249 |
| maximal normal subgroup of | #72225249 |
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, 3, F.5*F.8);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.9*F.11);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);