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 11149 conjugacy classes of subgroups; only the 3641 normal subgroups are shown.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(11,15)(13,16)(17,18)(19,22)(20,21)(23,24);(2,4)(18,21)(22,23);(1,2)(3,4)(6,8)(9,10,12,14)(11,16,15,13);(2,4)(5,6)(7,8)(17,19)(18,22)(20,24)(21,23);(5,6)(7,8)(9,10)(11,16)(12,14)(13,15);(1,2)(3,4)(9,11)(10,13)(12,15)(14,16)(17,18)(19,23)(20,21)(22,24)Polycyclic presentation:
2^10|c^2=j,b^a=bg,c^b=cgh,d^c=di,e^c=eghij,f^a=fj,f^b=fh,f^c=fj,f^d=fh,f^e=fj136 conjugacy classes — view the full character table.
| maximal subgroup of | #72327309, #72329191, #72335688, … |
| maximal normal subgroup of | #72327309, #72329191, #72335688, … |
| maximal quotient of | #72336072, #72336451 |
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.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);