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 | #9495 |
| commutator | C22×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C22×#120 | C23 |
| prime radical | #35346 | C22 |
| LCS | G ▷ C22×C22⋊C4 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C22×#120 ▷ C24 ▷ C22 ▷ C1 | |
2018 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 2018 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,13)(3,16,7,15)(5,21,17,11)(6,25,19,26)(8,12,18,30)(9,22,14,20)(10,28,27,24)(23,29,32,31);(2,9,14,16)(3,17,4,20)(5,22)(10,26,12,27)(11,21,13,15)(18,28,19,32)(23,24)(25,31,29,30);(2,10,12,14)(3,18)(4,19)(5,23)(9,26,27,16)(11,30,31,13)(15,29,25,21)(17,20)(22,24)(28,32)Polycyclic presentation:
2^10|a^2=f,b^2=dfi,c^2=hij,d^2=i,b^a=bd,c^a=cghij,c^b=ce,d^a=dg,d^b=dgi,d^c=dh,e^a=ehij,e^b=ehij,e^d=ej,f^b=fgi,f^c=fij,f^d=fi,f^e=fj,g^a=gi,g^b=gi,g^c=gj,h^a=hj55 conjugacy classes — view the full character table.
| maximal subgroup of | #72227793, #72266244, #72266326, … |
| maximal normal subgroup of | #72227793, #72266244, #72266326, … |
| maximal quotient of | #72317162, #72317196 |
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, 1, F.6);;
gap> SetPower(coll, 2, F.4*F.6*F.9);;
gap> SetPower(coll, 3, F.8*F.9*F.10);;
gap> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);