Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #8541 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | C24×D4 | C23 |
| prime radical | #6499892 | C2 |
| LCS | G ▷ C23×D4 ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C24×D4 ▷ C24 ▷ C2 ▷ C1 | |
3493 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 3493 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,20)(18,22)(19,21)(23,24);(5,8)(10,14)(15,20,22,24)(16,21)(17,23);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,17)(13,18)(14,20)(16,22)(19,23)(21,24)Polycyclic presentation:
2^10|b^2=f,b^a=bd,c^b=ce,d^b=dh,d^c=dg,e^a=egj,e^b=ei,e^d=ej,f^a=fh,f^c=fi,h^c=hj,i^a=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72309519, #72309892, #72309896 |
| maximal normal subgroup of | #72309519, #72309892, #72309896 |
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.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);