Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
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 12297 conjugacy classes of subgroups; only the 3993 normal subgroups are shown.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(5,6)(7,8)(14,17)(18,20);(5,6,7,8)(10,12)(14,17)(18,20);(2,4)(6,8);(10,12)(15,19)(18,20);(1,2,3,4)(5,6,7,8)(9,10)(11,12)(13,14,16,17)(15,18,19,20);(1,2)(3,4)(5,6)(7,8)(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^10|b^2=h,e^2=ij,b^a=bh,c^a=ch,c^b=ch,e^a=ehj,e^b=eg,e^c=ei,e^d=egj,f^b=fh,f^c=fi,f^d=fj,f^e=fi148 conjugacy classes — view the full character table.
| maximal subgroup of | #72263129, #72298061, #72300457, … |
| maximal normal subgroup of | #72263129, #72298061, #72300457, … |
| maximal quotient of | #72298568, #72298576, #72323725, … |
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.8);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);