Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #72262841 |
| commutator | C2×#535 | C22×C4 |
| Frattini, cosocle | #35527 | C23 |
| prime radical | #72318806 | C2 |
| LCS | G ▷ C2×#535 ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35527 ◁ G | |
| LECS | G ▷ #35527 ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
3151 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 3151 conjugacy classes of subgroups — too many to draw. Only its 94 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:
(1,2,5,9)(3,11)(4,17)(6,10,22,23)(7,16)(8,25)(12,28,27,31)(13,30,19,29)(14,24)(15,26)(18,20)(21,32);(1,2)(3,12)(4,18)(5,9)(6,23,22,10)(7,13,11,19)(8,21)(14,29,15,30)(16,27)(17,25,32,20)(24,31)(26,28);(1,3)(2,7)(4,19,21,13)(5,16)(6,14)(8,12,18,27)(9,11)(10,26)(15,22)(17,30,32,29)(20,31,25,28)(23,24)Polycyclic presentation:
2^11|a^2=hjk,b^2=f,c^2=h,d^2=i,g^2=k,b^a=be,c^a=ci,c^b=cd,d^a=dg,d^b=dh,d^c=dij,e^a=ej,e^b=ehj,e^c=egik,e^d=eik,f^a=fhj,f^c=fhi,f^d=fj,f^e=fj,g^b=gi,g^c=gk,g^e=gk,g^f=gk,h^b=hj,h^d=hk,i^a=ik,i^b=ik,j^c=jk56 conjugacy classes — view the full character table.
| maximal subgroup of | #72339082 |
| maximal normal subgroup of | #72339082 |
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.8*F.10*F.11);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 7, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
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.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);