Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #12759 |
| commutator | #127 | C2×C42 |
| Frattini, cosocle | C2×#532 | C23 |
| socle | C2 | #146599 |
| prime radical | C2×#3018 | C22 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×#532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
2173 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,8)(4,9,18,31)(5,10)(6,12)(7,11)(13,37,40,51)(14,42)(15,26)(16,28)(17,30,46,55)(19,32,20,33)(21,34)(22,35)(23,36)(24,39,29,41)(25,44)(27,45)(38,60,62,59)(43,53)(47,56,49,58)(48,57)(50,64,54,61)(52,63);(1,3,6,16)(2,8,12,28)(4,13,20,39)(5,14)(7,15)(9,29,33,37)(10,25)(11,27)(17,38,49,64)(18,40,19,41)(21,44)(22,43,48,63)(23,45)(24,32,51,31)(26,36)(30,54,58,60)(34,42)(35,53,57,52)(46,62,47,61)(50,56,59,55);(1,4,5,17)(2,9,10,30)(3,13,14,38)(6,18,21,46)(7,20,22,47)(8,24,25,50)(11,33,35,56)(12,31,34,55)(15,39,43,61)(16,40,44,62)(19,48,49,23)(26,37,52,59)(27,51,53,60)(28,29,42,54)(32,57,58,36)(41,63,64,45)Polycyclic presentation:
2^10|a^2=f,b^2=gj,c^2=d,e^2=i,h^2=j,b^a=bfg,c^a=ce,c^b=cehi,d^a=di,d^b=dij,e^a=eij,e^b=eh,f^c=fj,g^c=gh,g^d=gj,g^e=gi,h^a=hi,h^b=hij,h^g=hj,i^b=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72284997, #72286575, #72287090 |
| maximal normal subgroup of | #72284997, #72286575, #72287090 |
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.6);;
gap> SetPower(coll, 2, F.7*F.10);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);