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 | #411575 |
| commutator, Frattini, cosocle | C22×C4×C8 | C24 |
| LCS | G ▷ C22×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ C2×#21683 ◁ G | |
| LECS | G ▷ C22×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
5143 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 48 points:
(6,14)(9,15)(19,27,29,40)(20,31)(21,34)(23,35,36,48)(25,38)(26,37)(28,39,42,43)(30,45,44,41)(32,46)(33,47);(17,18)(19,28,29,42)(20,32)(21,26)(22,24)(23,30,36,44)(25,33)(27,39,40,43)(31,46)(34,37)(35,45,48,41)(38,47);(1,2)(3,9)(4,12)(5,8)(6,10)(7,13)(11,15)(14,16)(23,36)(28,42)(35,48)(39,43);(1,3)(2,6,8,14)(4,10)(5,11)(7,9,12,15)(13,16)(17,19,20,27,21,30,31,41,22,29,33,40,34,44,47,45)(18,23,25,35,26,28,38,43,24,36,32,48,37,42,46,39)Polycyclic presentation:
2^11|a^2=h,b^2=h,d^2=g,e^2=i,g^2=j,j^2=k,c^a=cf,c^b=ch,d^a=dg,d^b=dj,d^c=de,e^b=ek,e^c=ei,f^d=fi,g^a=gj,g^b=gk,g^c=gi,h^d=hk,j^a=jk116 conjugacy classes — view the full character table.
| maximal subgroup of | #72307289 |
| maximal normal subgroup of | #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, 4, F.7);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);