Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #72225440 |
| commutator, Frattini, cosocle | C8×C32 | C23 |
| prime radical | #72225422 | C2 |
| LCS | G ▷ C8×C32 ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C8×C16 ◁ #93664 ◁ G | |
| LECS | G ▷ C8×C32 ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
287 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 287 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,7,15)(3,8,18,33)(4,22,25,9)(5,26,28,11)(6,13,27,16)(10,40,43,17)(12,46,48,19)(14,49,45,23)(20,50,54,29)(21,31,55,34)(24,39,56,37)(30,61,35,52)(32,63,36,51)(38,57,47,60)(41,59,44,53)(42,64,62,58);(2,8)(3,17)(4,19)(5,23)(6,27)(9,35)(10,12)(11,36)(13,34)(14,50)(15,33)(16,31)(18,40)(20,58)(21,59)(22,30)(24,60)(25,46)(26,32)(28,49)(29,45)(37,42)(38,41)(39,62)(43,48)(44,47)(51,52)(53,55)(54,64)(56,57)(61,63);(2,9,14,38,13,37,11,12,15,22,45,47,16,39,26,48)(3,18)(8,10,32,42,31,41,29,35,33,43,36,62,34,44,50,30)(17,40)(20,54)(21,55)(51,63)(52,61)(53,59)(58,64)Polycyclic presentation:
2^11|a^2=k,c^2=egi,d^2=fh,e^2=gi,f^2=hj,g^2=i,h^2=jk,j^2=k,b^a=bd,c^a=cfk,c^b=ce,d^a=df,d^b=df,d^c=dk,e^a=ehk,e^b=eg,f^a=fh,f^b=fh,g^a=gj,g^b=gi,h^a=hj,h^b=hj,i^a=ik,j^a=jk,j^b=jk131 conjugacy classes — view the full character table.
| maximal subgroup of | #72225451 |
| maximal normal subgroup of | #72225451 |
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, 3, F.5*F.7*F.9);;
gap> SetPower(coll, 4, F.6*F.8);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
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, 1, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);