Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #147771 |
| commutator | C2×#124 | C22×C4 |
| Frattini, cosocle | C22×#124 | C23 |
| prime radical | C2×#3545 | C22 |
| LCS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C23×C4 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C22×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
2169 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)(3,9)(4,20)(5,11)(6,31)(7,13)(8,14)(10,24)(12,30)(15,49)(16,36)(17,37)(18,48)(19,39)(21,56)(22,41)(23,43)(25,55)(26,42)(27,47)(28,59)(29,46)(32,60)(33,45)(34,38)(35,50)(40,52)(44,58)(51,64)(53,63)(54,62)(57,61);(1,3)(2,9)(4,21,26,54)(5,28,29,17)(6,32)(7,34,22,18)(8,19)(10,40,44,49)(11,37,46,59)(12,36)(13,48,41,38)(14,39)(15,23,52,33)(16,30)(20,35,42,61)(24,53,58,51)(25,50,27,57)(31,60)(43,63,45,64)(47,62,55,56);(1,4,22,25,8,27,7,26)(2,10,41,43,14,45,13,44)(3,15,34,52,19,53,18,51)(5,23,6,33,29,58,30,24)(9,35,48,61,39,62,38,56)(11,42,31,20,46,55,12,47)(16,21,17,54,32,57,28,50)(36,40,59,49,60,64,37,63)Polycyclic presentation:
2^10|b^2=g,c^2=hj,d^2=i,e^2=ij,h^2=j,b^a=bf,c^a=ce,c^b=cd,d^b=dh,e^a=eij,e^b=ehi,e^c=ej,f^c=fhi,f^d=fij,f^e=fi,g^c=ghij,g^d=gij,g^e=gi,h^a=hi,h^b=hi,h^f=hj,h^g=hj,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72286182, #72286511, #72286670 |
| maximal normal subgroup of | #72286182, #72286511, #72286670 |
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, 2, F.7);;
gap> SetPower(coll, 3, F.8*F.10);;
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.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9*F.10);;
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);;
gap> SetConjugate(coll, 8, 6, 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);