Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #405081 |
| commutator, Frattini, cosocle | C22×C4×C8 | C24 |
| prime radical | #72264932 | C2 |
| LCS | G ▷ C22×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ C2×#21492 ◁ G | |
| LECS | G ▷ C22×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
5383 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,9)(4,12)(5,8)(6,10)(7,13)(11,15)(14,16)(19,28,32,33)(21,36)(24,39,40,44)(26,41)(29,46,42,43)(30,35)(31,47,38,45)(37,48);(6,14)(9,15)(17,18)(19,29,32,42)(20,25)(21,37)(22,34)(23,27)(24,31,40,38)(26,30)(28,43,33,46)(35,41)(36,48)(39,45,44,47);(1,3)(2,6,8,14)(4,10)(5,11)(7,9,12,15)(13,16)(17,19,30,45,20,28,21,38,23,32,35,47,22,33,36,31)(18,24,37,46,34,44,26,42,27,40,48,43,25,39,41,29);(17,18)(19,24)(20,34)(21,26)(22,25)(23,27)(28,44)(29,31)(30,37)(32,40)(33,39)(35,48)(36,41)(38,42)(43,47)(45,46)Polycyclic presentation:
2^11|a^2=h,b^2=h,c^2=fhj,e^2=ik,f^2=j,j^2=k,b^a=bg,c^a=ce,c^b=cf,d^a=dh,e^a=ei,e^d=ek,f^a=fik,f^b=fj,f^c=fk,g^c=gik,h^c=hk,j^b=jk122 conjugacy classes — view the full character table.
| maximal subgroup of | #72265173, #72307289 |
| maximal normal subgroup of | #72265173, #72307289 |
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);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.6*F.8*F.10);;
gap> SetPower(coll, 5, F.9*F.11);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
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.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);