Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C26 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
102293 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(13,14,17,18)(15,19,20,16);(9,10)(11,12);(5,6)(7,8)(10,12)(15,20)(16,19);(2,4)(6,7)(14,18)(16,19);(9,10)(11,12)(13,15)(14,19)(16,18)(17,20);(1,2,3,4)(5,7,8,6)(9,10)(11,12)(13,16,17,19)(14,15,18,20)Polycyclic presentation:
2^10|a^2=g,f^2=i,c^b=cgh,d^a=dg,d^c=dj,e^c=eh,f^c=fhj,f^d=fi184 conjugacy classes — view the full character table.
| maximal subgroup of | #72234056, #72237367, #72237469, … |
| maximal normal subgroup of | #72234056, #72237367, #72237469, … |
| maximal quotient of | #72302232, #72302256, #72330675, … |
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.7);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);