Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #147824 |
| commutator | #532 | C22×C4 |
| Frattini, cosocle | C2×#532 | C23 |
| prime radical | #7500918 | C2 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×#532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
2228 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)(4,18,21,49)(5,24)(6,28)(7,11)(9,20,35,53)(10,26)(12,29)(13,41,43,48)(14,31)(15,40)(16,32)(17,33)(19,51,44,64)(22,50,23,56)(25,38)(27,39)(30,42,61,45)(34,59,62,52)(36,54,37,55)(46,57)(47,63)(58,60);(1,3)(2,8)(4,19,22,13)(5,25,26,46)(6,15,29,16)(7,17)(9,34,36,61)(10,38,24,57)(11,33)(12,40,28,32)(14,27)(18,48,50,64)(20,42,54,52)(21,44,23,43)(30,35,62,37)(31,39)(41,56,51,49)(45,55,59,53)(47,60)(58,63);(1,4)(2,9)(3,13)(5,20)(6,22)(7,21)(8,30)(10,18)(11,35)(12,36)(14,45)(15,44)(16,19)(17,43)(23,29)(24,49)(25,59)(26,53)(27,55)(28,37)(31,48)(32,62)(33,61)(34,40)(38,64)(39,56)(41,63)(42,47)(46,52)(50,58)(51,57)(54,60)Polycyclic presentation:
2^10|a^2=f,b^2=g,d^2=ij,e^2=ij,h^2=j,b^a=bfg,c^a=cd,c^b=ce,d^a=di,d^b=dh,d^c=dij,e^a=ej,e^b=ehi,e^c=eij,f^c=fj,g^c=gh,g^d=gi,g^e=gij,h^a=hi,h^b=hij,h^c=hj,h^g=hj,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72286575, #72286910, #72287068 |
| maximal normal subgroup of | #72286575, #72286910, #72287068 |
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);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetPower(coll, 5, F.9*F.10);;
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.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
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.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*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);