Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#442 |
| commutator | C2×C8 | C24×C4 |
| Frattini, cosocle | C4×C8 | C25 |
| prime radical | #7623517 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ C22×#322 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
2466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,7)(3,8,11,16)(4,12,13,9)(5,10,15,14)(17,18)(19,25)(20,26)(21,37)(22,28)(23,27)(24,29)(30,39)(31,42)(32,41)(33,43)(34,44)(35,45)(36,46)(38,40)(47,48);(3,11)(4,13)(8,16)(9,12)(17,19)(18,25)(20,34)(21,38)(22,32)(23,31)(24,33)(26,44)(27,42)(28,41)(29,43)(30,36)(35,47)(37,40)(39,46)(45,48);(1,3,2,8,6,11,7,16)(4,5,12,10,13,15,9,14)(17,20,23,35,24,36,21,22)(18,26,27,45,29,46,37,28)(19,30,31,32,33,34,38,47)(25,39,42,41,43,44,40,48);(1,4,5,8,7,12,14,11,6,13,15,16,2,9,10,3)(17,21,24,23)(18,27,29,37)(19,31,33,38)(20,22,36,35)(25,40,43,42)(26,45,46,28)(30,32,34,47)(39,48,44,41);(1,2,6,7)(3,11)(5,14,15,10)(8,16)(17,22,21,36,24,35,23,20)(18,28,37,46,29,45,27,26)(19,32,38,30,33,47,31,34)(25,41,40,39,43,48,42,44)Polycyclic presentation:
2^10|a^2=j,c^2=f,d^2=ghj,e^2=fhi,f^2=h,g^2=i,i^2=j,c^b=ch,d^a=dh,d^b=dh,d^c=dg,e^c=eij,e^d=eij,f^d=fj,g^c=gi,g^e=gj,i^c=ij112 conjugacy classes — view the full character table.
| maximal subgroup of | #72287479, #72287589, #72287605, … |
| maximal normal subgroup of | #72287479, #72287589, #72287605, … |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.8*F.10);;
gap> SetPower(coll, 5, F.6*F.8*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);