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 | #7624085 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ #17320 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
2201 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(17,18,23,31,44,47,22,43,25,33,45,29,24,32,37,30)(19,26,35,40,48,42,34,39,36,21,41,20,27,38,28,46);(1,2)(3,6)(4,9)(5,14)(7,11)(8,13)(10,15)(12,16)(17,19)(18,26)(20,29)(21,33)(22,34)(23,35)(24,27)(25,36)(28,37)(30,46)(31,40)(32,38)(39,43)(41,45)(42,47)(44,48);(1,3)(2,8)(4,14)(5,12)(6,11)(7,13)(9,15)(10,16)(17,20,37,21,24,39,45,42,25,40,22,26,44,46,23,38)(18,27,30,41,32,36,29,34,33,48,43,35,47,19,31,28);(1,4,6,15,7,16,8,5)(2,9,3,10,11,12,13,14)(17,21,22,39,24,42,23,40,25,26,37,46,44,38,45,20)(18,28,30,48,32,41,29,19,33,34,43,27,47,35,31,36);(1,5,8,16,7,15,6,4)(2,10,13,9,11,14,3,12)(18,29,33,31)(20,21,40,26)(22,23)(24,44)(27,48)(28,41)(30,47,43,32)(34,35)(37,45)(38,46,42,39)Polycyclic presentation:
2^10|a^2=g,c^2=gi,d^2=f,e^2=fghi,f^2=h,g^2=ij,h^2=j,i^2=j,c^b=chi,d^a=di,d^c=dh,e^a=egij,e^b=ehij,e^c=eg,e^d=egij,f^a=fj,f^c=fj,f^e=fi,g^d=gj,g^e=gij,h^e=hj,i^e=ij103 conjugacy classes — view the full character table.
| maximal subgroup of | #72318603, #72318841, #72318898, … |
| maximal normal subgroup of | #72318603, #72318841, #72318898, … |
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.7);;
gap> SetPower(coll, 3, F.7*F.9);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.6*F.7*F.8*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 5, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);