Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 11007 conjugacy classes of subgroups; only the 3637 normal subgroups are shown.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(6,8)(20,24)(22,23);(1,2,3,4)(6,8)(17,18)(19,21)(20,23)(22,24);(6,8)(11,15)(14,16)(20,24)(22,23);(10,13)(14,16)(18,19)(22,23);(5,6)(7,8)(9,10,12,13)(11,16,15,14)(17,19,21,18)(20,23,24,22);(1,2)(3,4)(9,11,12,15)(10,14,13,16)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^10|b^2=h,e^2=i,f^2=ij,b^a=bh,d^b=dj,e^a=eg,e^b=egj,e^c=eg,e^d=ei,f^a=fhj,f^b=fhj,f^c=fi,f^e=fij136 conjugacy classes — view the full character table.
| maximal subgroup of | #72234830, #72263129, #72331604, … |
| maximal normal subgroup of | #72234830, #72263129, #72331604, … |
| maximal quotient of | #72333335, #72333338 |
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);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);