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 | |
156039 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,12,13)(11,15,16,14)(18,21)(22,24);(2,4)(6,8)(17,18)(19,22)(20,21)(23,24);(2,4)(6,8)(19,23)(22,24);(1,2,3,4)(5,6,7,8)(10,13)(14,15)(18,21)(22,24);(2,4)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^10|a^2=g,d^2=gh,b^a=bj,c^a=cgh,d^a=dh,d^b=dj,d^c=dgh,e^a=ehi,e^d=ehi,f^a=fg,f^b=fgh,f^c=fj,f^d=fgh,f^e=fij160 conjugacy classes — view the full character table.
| maximal subgroup of | #72256567, #72268592, C2×#72241007, … |
| maximal normal subgroup of | #72256567, #72268592, C2×#72241007, … |
| maximal quotient of | #72296166, C2×#72241007, #72323740, … |
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, 4, F.7*F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);