Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
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 | |
At least 13183 conjugacy classes of subgroups; only the 4505 normal subgroups are shown.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2)(3,4)(5,6)(7,8)(10,12);(1,2,4,3)(5,6,8,7)(9,10,11,12)(13,14)(15,18)(16,19)(17,20);(1,2,4,3)(5,6,8,7)(10,12)(13,15)(14,17)(16,20)(18,19);(2,3)(6,7)(9,10)(11,12);(6,7)(13,15)(14,18)(16,20)(17,19);(1,3,4,2)(5,7,8,6)(13,14)(15,17)(16,19)(18,20)Polycyclic presentation:
2^10|b^2=j,c^2=i,f^2=i,b^a=bj,c^a=ci,c^b=cg,d^a=dj,d^b=dj,d^c=dj,e^a=eh,e^b=eh,e^c=eh,f^a=fi,f^d=fi,f^e=fghij160 conjugacy classes — view the full character table.
| maximal subgroup of | #72232191, #72232274, #72232438, … |
| maximal normal subgroup of | #72232191, #72232274, #72232438, … |
| maximal quotient of | #72252093, #72278177, #72278229, … |
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.10);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);