Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #10078 |
| commutator, Frattini, cosocle | C2×#376 | C23 |
| socle | C2 | #102906 |
| prime radical | #233619 | C2 |
| LCS | G ▷ C2×#376 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C2×#376 ▷ C22×C4 ▷ C2 ▷ C1 | |
1523 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,8,13)(3,9,18,36)(4,19,25,43)(5,11,6,28)(7,12,32,46)(10,27,41,22)(14,47,51,62)(15,52,16,34)(17,35,54,63)(20,57,58,61)(21,44,30,38)(23,45,59,40)(24,39,31,56)(26,60,29,42)(33,53,55,48)(37,50,64,49);(1,3)(2,9)(4,20,24,53,25,58,31,48)(5,26,6,29)(7,17)(8,18)(10,37,40,47,41,64,45,62)(11,42,28,60)(12,35)(13,36)(14,23,49,27,51,59,50,22)(15,21,16,30)(19,55,56,57,43,33,39,61)(32,54)(34,44,52,38)(46,63);(1,4,21,59,8,25,30,23)(2,10,38,56,13,41,44,39)(3,14,26,20)(5,27,7,31,6,22,32,24)(9,33,60,37)(11,43,46,40,28,19,12,45)(15,53,54,50)(16,48,17,49)(18,51,29,58)(34,62,35,61)(36,55,42,64)(47,63,57,52)Polycyclic presentation:
2^10|a^2=j,b^2=f,c^2=efghij,d^2=h,e^2=i,f^2=hij,g^2=j,b^a=bf,c^a=cd,c^b=ce,d^a=dh,d^b=dg,d^c=dj,e^a=ei,e^b=eg,e^c=egi,e^d=ej,f^a=fhij,f^c=fgij,f^d=fj,f^e=fj,g^c=gj,h^b=hj,i^b=ij,i^c=ij70 conjugacy classes — view the full character table.
| maximal subgroup of | #72279769, #72279936, #72280230, … |
| maximal normal subgroup of | #72279769, #72279936, #72280230, … |
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.10);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.5*F.6*F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.8*F.9*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
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.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);