Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #511530 |
| commutator, Frattini, cosocle | C22×C16 | C24 |
| prime radical | C2×#2263 | C22 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2960 ◁ #33475 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
559 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,3,9,16)(4,10,21,35,24,14,23,19)(5,11,27,38,28,15,30,42)(7,13,26,37,29,17,31,40)(18,34,45,43,48,32,47,49)(20,33,53,50,25,39,54,61)(22,36,55,58,57,41,44,51)(46,60,64,62,52,59,56,63);(2,3)(4,18)(5,25)(6,9)(7,26)(10,32)(11,33)(13,40)(14,34)(15,39)(17,37)(19,49)(20,28)(21,47)(22,56)(23,45)(24,48)(27,53)(29,31)(30,54)(35,43)(36,62)(38,61)(41,63)(42,50)(44,46)(51,59)(52,55)(57,64)(58,60);(1,3,6,16,8,2,9,12)(4,19,21,10,24,35,23,14)(5,11,27,38,28,15,30,42)(7,17,26,40,29,13,31,37)(18,43,45,32,48,49,47,34)(20,33,53,50,25,39,54,61)(22,51,55,36,57,58,44,41)(46,62,64,59,52,63,56,60);(1,4,20,52,7,22,27,47,9,21,54,56,31,55,5,18,8,24,25,46,29,57,30,45,6,23,53,64,26,44,28,48)(2,10,33,59,13,36,38,49,16,35,61,63,40,58,11,34,3,14,39,60,17,41,42,43,12,19,50,62,37,51,15,32)Polycyclic presentation:
2^10|a^2=g,c^2=g,d^2=ehj,e^2=hi,g^2=j,h^2=ij,i^2=j,b^a=bf,c^a=cgij,c^b=cf,d^a=dfj,d^b=de,d^c=dij,e^b=eh,e^c=ej,f^a=fj,f^c=fj,g^b=gj,g^d=gj,h^b=hi,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72321482 |
| maximal normal subgroup of | #72321482 |
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);;
gap> SetPower(coll, 4, F.5*F.8*F.10);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);