Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C26 |
| prime radical | C22×#560 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
11133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(19,21)(20,24)(26,31)(27,29);(3,7)(5,8)(10,13)(14,16)(18,23)(19,21)(26,31)(28,32);(17,18)(19,20)(21,24)(22,23)(25,26,30,31)(27,32,29,28);(17,19,22,21)(18,20,23,24)(25,27,30,29)(26,32,31,28);(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,24)(18,19,23,21)(25,28)(26,27)(29,31)(30,32);(1,3,4,7)(2,5,6,8)(9,11,12,15)(10,14,13,16)(17,21,22,19)(18,24,23,20)(25,29,30,27)(26,28,31,32)Polycyclic presentation:
2^10|c^2=g,d^2=h,e^2=ij,f^2=j,c^a=cg,c^b=ch,d^a=dh,d^b=dgh,e^a=egh,e^b=ei,f^a=fh,f^b=fij160 conjugacy classes — view the full character table.
| Fitting subgroup of | #72258805, #72258832, #72258879, … |
| maximal subgroup of | #72258805, #72258832, #72258879, … |
| maximal normal subgroup of | #72258805, #72258832, #72258879, … |
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, 3, F.7);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
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.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);