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 | |
58645 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(5,6)(7,8)(9,10)(11,13)(12,14)(15,16);(2,4)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,21)(22,24);(1,2,3,4)(5,7,8,6)(11,15)(13,16)(18,21)(22,23);(9,10,12,14)(11,16,15,13)(17,19,20,24)(18,22,21,23);(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,24)(21,22);(9,11)(10,13)(12,15)(14,16)(17,18)(19,23)(20,21)(22,24)Polycyclic presentation:
2^10|c^2=g,d^2=j,b^a=bh,c^a=ch,c^b=chi,d^a=dghij,d^b=dj,e^d=ej,f^c=fj,f^d=fj172 conjugacy classes — view the full character table.
| maximal subgroup of | #72259088, #72266385, #72266546, … |
| maximal normal subgroup of | #72259088, #72266385, #72266546, … |
| maximal quotient of | #72326997, #72330598, #72330660, … |
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, 3, F.7);;
gap> SetPower(coll, 4, 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*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);