Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×C22⋊C4 |
| commutator | C24 | C24×C4 |
| Frattini, cosocle | C25 | C25 |
| socle | C23 | C23×C22⋊C4 |
| prime radical | C25×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C26×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
57957 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^10|a^2=i,c^2=f,c^a=cg,c^b=cgi,d^a=dj,d^b=dj,d^c=dh,e^c=ei,f^d=fj,h^c=hj208 conjugacy classes — view the full character table.
| maximal subgroup of | #72236641, #72237685, #72238028, … |
| maximal normal subgroup of | #72236641, #72237685, #72238028, … |
| maximal quotient of | #72261682, #72264322, #72293120, … |
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.9);;
gap> SetPower(coll, 3, F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);