Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #148237 |
| commutator | C2×#124 | C22×C4 |
| Frattini, cosocle | C22×#124 | C23 |
| prime radical | C2×#3153 | 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 | |
1453 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)(4,20,24,45)(5,26)(6,12)(7,13)(8,14)(10,23,42,33)(11,30)(15,49,51,64)(16,54)(17,56)(18,47)(19,39)(21,59,55,62)(22,60,28,43)(25,44,61,41)(27,37)(29,48)(31,46)(32,36)(34,38)(35,50,58,57)(40,52,63,53);(1,3)(2,9)(4,21,25,50)(5,27,30,17)(6,32)(7,34,29,18)(8,19)(10,40,43,64)(11,37,26,56)(12,36)(13,47,48,38)(14,39)(15,22,52,23)(16,31)(20,58,44,59)(24,55,61,57)(28,53,33,51)(35,41,62,45)(42,63,60,49)(46,54);(1,4,7,25,8,24,29,61)(2,10,13,43,14,42,48,60)(3,15,34,52,19,51,18,53)(5,28,31,23,30,22,6,33)(9,35,47,62,39,58,38,59)(11,44,12,45,26,41,46,20)(16,55,17,57,32,21,27,50)(36,63,56,49,54,40,37,64)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=cd,d^a=dj,d^b=dh,d^c=dj,e^a=ei,e^b=ehi,e^c=ej,f^c=fj,g^c=ghij,g^d=gij,g^e=gi,h^a=hi,h^b=hi,h^g=hj,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72286360, #72286511, #72286616 |
| maximal normal subgroup of | #72286360, #72286511, #72286616 |
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.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
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.8*F.9);;
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*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, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);