Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C27 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
At least 8945 conjugacy classes of subgroups; only the 39424 normal subgroups are shown.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 24 points:
(1,2)(3,4)(5,6)(7,8);(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,14)(13,16);(5,6)(7,8)(9,10,12,14)(11,13,16,15);(17,18)(19,22)(20,23)(21,24);(1,2,3,4)(5,6,7,8)(17,19)(18,20)(21,23)(22,24);(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,24)(21,23);(1,2)(3,4)(5,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,20,21,22)(18,23,24,19)Polycyclic presentation:
2^11|c^2=i,e^2=j,g^2=k,c^b=ci,e^a=ej,e^b=ej,e^c=eh,e^d=ek,f^c=fi,g^b=gi,g^e=gj,g^f=gk308 conjugacy classes — view the full character table.
| maximal subgroup of | #72237495, #72249440, #72249566, … |
| maximal normal subgroup of | #72237495, #72249440, #72249566, … |
| maximal quotient of | #72330739, #72335028, #72335072, … |
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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 7, F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);