Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #10336 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | C23×#41 | C23 |
| socle | C2 | #116351 |
| prime radical | #62811 | C22 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×#41 ▷ C23 ▷ C2 ▷ C1 | |
2921 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,10,7,12,22,34)(3,8,14,26,16,28,40,52)(4,9,18,30,20,32,44,56)(6,11,21,33,24,36,48,59)(13,25,37,49,38,50,60,63)(15,35,39,58,23,27,47,51)(17,29,41,53,42,54,61,64)(19,45,43,62,46,31,57,55);(1,3)(2,8)(4,17,20,42)(5,14)(6,23,24,15)(7,16)(9,29,32,54)(10,26)(11,35,36,27)(12,28)(13,19)(18,41,44,61)(21,47,48,39)(22,40)(25,31)(30,53,56,64)(33,58,59,51)(34,52)(37,43)(38,46)(45,50)(49,55)(57,60)(62,63);(1,4)(2,9)(3,13,16,38)(5,18)(6,19)(7,20)(8,25,28,50)(10,30)(11,31)(12,32)(14,37,40,60)(15,17,23,42)(21,43)(22,44)(24,46)(26,49,52,63)(27,29,35,54)(33,55)(34,56)(36,45)(39,41,47,61)(48,57)(51,53,58,64)(59,62)Polycyclic presentation:
2^10|a^2=d,b^2=fhi,c^2=h,d^2=j,e^2=j,b^a=bf,c^a=cg,c^b=ce,e^a=ei,e^b=eh,f^c=fh,f^e=fj,g^b=ghi,g^e=gj,h^b=hj,i^b=ij,i^c=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72342838, #72342843, #72342848, … |
| maximal normal subgroup of | #72342838, #72342843, #72342848, … |
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.4);;
gap> SetPower(coll, 2, F.6*F.8*F.9);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 5, 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);