Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#13424 |
| commutator, Frattini, cosocle | C2×#124 | C24 |
| prime radical | C22×#454 | C22 |
| LCS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C23×C4 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1891 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)(3,9)(4,20,24,44)(5,27)(6,12)(7,13)(8,14)(10,23,41,33)(11,30)(15,47,51,63)(16,54)(17,55)(18,57)(19,38)(21,59,49,62)(22,43,60,42)(25,58,26,40)(28,36)(29,46)(31,45)(32,35)(34,50,61,56)(37,48)(39,52,64,53);(1,3)(2,9)(4,21,25,56)(5,28,30,17)(6,32)(7,18,29,48)(8,19)(10,39,42,63)(11,36,27,55)(12,35)(13,37,46,57)(14,38)(15,22,52,23)(16,31)(20,34,58,59)(24,49,26,50)(33,51,60,53)(40,62,44,61)(41,64,43,47)(45,54);(1,4,7,26,8,24,29,25)(2,10,13,43,14,41,46,42)(3,15,48,52,19,51,18,53)(5,22,31,23,30,60,6,33)(9,34,57,59,38,61,37,62)(11,40,12,44,27,58,45,20)(16,49,28,50,32,21,17,56)(35,39,36,63,54,64,55,47);(1,5)(2,11)(3,16,19,32)(4,22)(6,29)(7,31)(8,30)(9,35,38,54)(10,40)(12,13)(14,27)(15,49,51,21)(17,48,28,18)(20,42)(23,26)(24,60)(25,33)(34,39,61,64)(36,37,55,57)(41,58)(43,44)(45,46)(47,59,63,62)(50,53,56,52)Polycyclic presentation:
2^10|a^2=f,b^2=g,c^2=hi,d^2=i,e^2=ij,h^2=j,b^a=bg,c^a=ce,c^b=cefh,d^a=dj,d^b=dhi,e^a=ei,e^b=eh,e^c=ej,f^c=fj,g^c=gh,g^d=gij,g^e=gi,h^a=hi,h^b=hij,h^g=hj,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72286240, #72286493, #72286584 |
| maximal normal subgroup of | #72286240, #72286493, #72286584 |
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, 3, F.8*F.9);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*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*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);