Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #72216452 |
| commutator | #1781 | C22×C4 |
| Frattini, cosocle | #20586 | C23 |
| prime radical | #72241386 | C2 |
| LCS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G | |
| LECS | G ▷ #20586 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
2648 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 2648 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10)(3,14,17,19)(4,16,18,31)(6,11,24,28)(7,12,22,27)(8,13,23,29)(9,25,26,15)(20,30,21,32);(1,3,15,30,8,11,7,16)(2,4,12,19,13,20,9,6)(5,17,25,32,23,28,22,31)(10,18,27,14,29,21,26,24);(1,4)(2,9)(3,16)(5,21,23,18)(6,15)(7,19)(8,20)(10,27,29,26)(11,30)(12,13)(14,25,24,22)(17,31,28,32)Polycyclic presentation:
2^11|a^2=d,b^2=fgjk,c^2=hjk,e^2=h,b^a=bdf,c^a=ce,c^b=cegjk,d^c=dg,e^a=egh,e^b=ei,e^c=ehj,e^d=eijk,f^b=fik,f^c=fhjk,f^e=fj,g^a=gi,g^b=gi,g^c=gk,g^e=gjk,g^f=gk,h^a=hjk,h^b=hk,h^d=hk,i^b=ik,i^c=ij,i^e=ik,j^a=jk,j^b=jk62 conjugacy classes — view the full character table.
| maximal subgroup of | #72252357, #72284333, #72285547, … |
| maximal normal subgroup of | #72252357, #72285547, #72286010, … |
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.6*F.7*F.10*F.11);;
gap> SetPower(coll, 3, F.8*F.10*F.11);;
gap> SetPower(coll, 5, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.7*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 5, 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);