Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#7096 |
| commutator | C8×C16 | C22×C4 |
| Frattini, cosocle | C162 | C23 |
| prime radical | C2×#123403 | C2 |
| LCS | G ▷ C8×C16 ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ C2×C162 ◁ G | |
| LECS | G ▷ C162 ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
890 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 890 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 512 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,7)(4,16)(5,19)(6,8)(9,12)(10,13)(11,17)(14,26)(15,20)(18,32)(21,28)(22,27)(23,29)(24,30)(25,31)(33,34,35,41,36,40,46,55,50,42,59,57,62,54,61,64,38,44,48,58,51,53,60,63,37,43,47,56,49,39,45,52);(2,7)(3,12)(4,13)(5,17)(8,20)(9,10)(11,26)(14,31)(15,29)(16,21)(18,30)(19,22)(23,24)(25,32)(27,28)(34,39)(35,45)(36,49)(37,50)(40,43)(41,56)(42,53)(44,54)(46,47)(48,61)(51,62)(55,63)(57,58)(59,60);(2,8)(7,20)(9,23)(10,24)(11,19)(16,32)(21,25)(22,26)Polycyclic presentation:
2^11|a^2=e,d^2=fgh,e^2=g,f^2=hj,g^2=ik,h^2=j,i^2=k,b^a=bd,c^a=cjk,d^a=df,d^b=dfgi,e^b=egi,f^a=fh,f^b=fh,g^b=gi,h^a=hj,h^b=hj,i^b=ik182 conjugacy classes — view the full character table.
| maximal subgroup of | #72227058, #72227092, #72227114, … |
| maximal normal subgroup of | #72227058, #72227092, #72227114, … |
| maximal quotient of | #72227042 |
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.5);;
gap> SetPower(coll, 4, F.6*F.7*F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9*F.11);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.10*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);