Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #392667 |
| commutator | C22×C8 | C23×C4 |
| Frattini, cosocle | C23×C8 | C24 |
| prime radical | #6629044 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ C2×#3573 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
5414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,8,12,23)(5,22)(6,25)(9,24)(10,26,32,43)(11,27,18,46)(13,37,36,28)(14,38)(15,42)(17,45)(19,34)(21,49)(29,44)(30,48)(31,55,39,50)(33,59,58,51)(35,54,47,52)(40,53)(56,60)(57,63,62,64);(1,2,9,28,25,36,24,8,7,12,22,37,6,13,5,23)(3,10,29,51,45,58,44,26,16,32,42,59,17,33,15,43)(4,18,30,54,49,35,48,46,20,11,34,52,21,47,19,27)(14,39,53,64,61,57,60,55,38,31,56,63,41,62,40,50);(1,3,7,16)(2,10,12,32)(4,14,20,38)(5,15,24,44)(6,17,25,45)(8,26,23,43)(9,29,22,42)(11,31,18,39)(13,33,36,58)(19,40,48,60)(21,41,49,61)(27,50,46,55)(28,51,37,59)(30,53,34,56)(35,57,47,62)(52,63,54,64);(1,4,7,20)(2,11,12,18)(3,14,16,38)(5,19,24,48)(6,21,25,49)(8,27,23,46)(9,30,22,34)(10,31,32,39)(13,35,36,47)(15,40,44,60)(17,41,45,61)(26,50,43,55)(28,52,37,54)(29,53,42,56)(33,57,58,62)(51,63,59,64)Polycyclic presentation:
2^10|a^2=e,b^2=efij,c^2=j,d^2=j,f^2=i,i^2=j,b^a=bf,c^a=ch,d^a=dg,d^b=dj,e^b=ej,f^a=fi,f^b=fj,g^c=gj,h^d=hj,i^a=ij154 conjugacy classes — view the full character table.
| Fitting subgroup of | #72341798 |
| maximal subgroup of | #72320817, #72321054, #72341798, … |
| maximal normal subgroup of | #72320817, #72321054, #72341798, … |
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.5);;
gap> SetPower(coll, 2, F.5*F.6*F.9*F.10);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);