Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #10995 |
| commutator | C24×D4 | C22×C4 |
| Frattini, cosocle | C25×D4 | C23 |
| prime radical | #72218795 | C2 |
| LCS | G ▷ C24×D4 ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ C25×D4 ◁ G | |
| LECS | G ▷ C25×D4 ▷ C24 ▷ C2 ▷ C1 | |
At least 6833 conjugacy classes of subgroups; only the 272 normal subgroups are shown.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(9,10)(11,13)(12,14)(15,16)(21,24);(1,2)(3,7)(4,6)(5,8)(13,16)(18,21)(22,24);(1,3)(2,5,6,7)(4,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,21)(20,22)(23,24)Polycyclic presentation:
2^11|c^2=d,b^a=bg,c^a=cf,c^b=ce,d^b=di,e^a=eh,e^c=ei,f^b=fhj,f^e=fk,g^c=gjk,h^c=hk122 conjugacy classes — view the full character table.
| maximal subgroup of | #72258296, #72258588, #72265810, … |
| maximal normal subgroup of | #72258296, #72258588, #72265810, … |
| maximal quotient of | #72226375, #72266182, #72266294, … |
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.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);