Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
At least 12038 conjugacy classes of subgroups; only the 31797 normal subgroups are shown.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 20 points:
(5,6)(7,8)(9,11)(10,12);(2,4)(5,7,10,9)(6,11,12,8)(13,14)(15,16)(17,19)(18,20);(1,2,3,4)(13,14)(15,16)(17,19)(18,20);(5,8)(6,7)(9,12)(10,11)(13,15)(14,18)(16,19)(17,20);(5,6)(7,11)(8,9)(10,12)(13,14)(15,16)(17,19)(18,20);(5,6)(7,8)(9,11)(10,12)(13,14)(15,18)(16,20)(17,19);(5,9,10,7)(6,8,12,11)(13,16,17,18)(14,15,19,20)Polycyclic presentation:
2^10|b^2=i,c^2=h,g^2=j,b^a=bi,c^b=ch,d^b=dij,d^c=dij,e^d=ej,f^b=fi,g^a=gi,g^f=gj184 conjugacy classes — view the full character table.
| maximal subgroup of | #72262248, #72262626 |
| maximal normal subgroup of | #72262248, #72262626 |
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.9);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);