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 | #7624083 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ #17320 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
2208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,5)(4,9)(6,12)(7,10)(8,11)(13,16)(14,15)(17,18,22,26)(19,23,31,29)(20,24,37,35)(21,25,33,43)(27,40,47,46)(28,41,38,30)(32,42,45,48)(34,36,44,39);(1,3,7,13,8,4,6,14)(2,5,10,16,11,9,12,15)(17,19,18,23,22,31,26,29)(20,32,35,48,37,45,24,42)(21,38,25,30,33,28,43,41)(27,44,46,36,47,34,40,39);(1,4,7,14,8,3,6,13)(2,9,10,15,11,5,12,16)(17,20,33,36,18,24,43,44,22,37,21,39,26,35,25,34)(19,27,30,45,29,46,38,42,31,47,41,32,23,40,28,48);(1,5,7,16,8,9,6,15)(2,4,10,14,11,3,12,13)(17,21,26,43,22,33,18,25)(19,28,29,30,31,38,23,41)(20,34,24,36,37,44,35,39)(27,45,40,48,47,32,46,42);(1,6,8,7)(2,10,11,12)(3,13,4,14)(5,15,9,16)(17,18,22,26)(19,29,31,23)(20,24,37,35)(21,25,33,43)(27,46,47,40)(28,30,38,41)(32,48,45,42)(34,36,44,39)Polycyclic presentation:
2^10|a^2=j,b^2=f,c^2=fgh,d^2=fj,e^2=i,f^2=i,g^2=h,h^2=j,c^b=cg,d^a=dij,d^b=di,d^c=dhi,e^a=eij,e^b=ei,e^c=eij,f^c=fj,g^b=gh,g^c=gj,g^d=gj,h^b=hj115 conjugacy classes — view the full character table.
| maximal subgroup of | #72287387, #72287545, #72321717, … |
| maximal normal subgroup of | #72287387, #72287545, #72321717, … |
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, 2, F.6);;
gap> SetPower(coll, 3, F.6*F.7*F.8);;
gap> SetPower(coll, 4, F.6*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);