Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #72237744 |
| commutator, Frattini, cosocle | #1781 | C24 |
| prime radical | #72241347 | C2 |
| LCS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G | |
| LECS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
2710 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 2710 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 4 generators on 32 points:
(2,7)(5,18)(8,20)(9,21)(10,14)(16,29)(19,23)(22,31)(24,25)(30,32);(2,8)(5,18)(7,20)(9,22)(10,19)(14,23)(21,31)(24,30)(25,32)(26,27);(1,2,6,10)(3,9)(4,14)(5,19,18,8)(7,17)(11,25,15,21)(12,22)(13,24)(16,23)(20,29)(26,31,27,32)(28,30);(1,3)(2,9)(4,15,17,11)(5,12,18,28)(6,13)(7,21,14,25)(8,22,19,30)(10,24)(16,27)(20,31)(23,32)(26,29)Polycyclic presentation:
2^11|c^2=h,d^2=hi,f^2=h,c^a=ce,c^b=cf,d^a=dg,d^b=dg,d^c=dhijk,e^b=eg,e^c=ej,e^d=eijk,f^a=fgij,f^b=fh,f^c=fh,f^d=fik,f^e=fij,g^c=gi,g^d=gj,g^f=gjk,h^a=hj,h^d=hk,h^e=hk,i^b=ij,i^d=ik,i^f=ik,j^c=jk50 conjugacy classes — view the full character table.
| maximal subgroup of | #72285855, #72286010, #72286183, … |
| maximal normal subgroup of | #72285855, #72286010, #72286183, … |
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, 3, F.8);;
gap> SetPower(coll, 4, F.8*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.11);;
gap> SetConjugate(coll, 9, 6, F.9*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);