Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C44 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #8546 |
| commutator | C22×C42 | C22×C4 |
| Frattini, cosocle | C22×#127 | C23 |
| prime radical | #6342426 | C2 |
| LCS | G ▷ C22×C42 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#127 ◁ G | |
| LECS | G ▷ C22×#127 ▷ C24 ▷ C22 ▷ C1 | |
7153 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 7153 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^10|a^2=h,b^2=g,c^2=d,e^2=i,f^2=j,c^a=cf,c^b=ce,d^a=dh,d^b=dg,e^c=egi,e^d=ei,f^c=fhj,f^d=fj,g^c=gi,h^c=hj88 conjugacy classes — view the full character table.
| maximal subgroup of | #72210019, #72210020, #72210021, … |
| maximal normal subgroup of | #72210019, #72210059, #72219745, … |
| maximal quotient of | C2×#72210012, #72266359, #72266398, … |
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.8);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);