Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#2403 |
| commutator, Frattini, cosocle | C23×C4 | C25 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
43753 conjugacy classes of subgroups.
Polycyclic presentation:
2^10|c^2=g,g^2=j,c^a=cg,c^b=cf,d^b=di,e^c=eh,f^d=fj,g^a=gj,i^c=ij124 conjugacy classes — view the full character table.
| maximal subgroup of | #72214572, #72216762, #72231759, … |
| maximal normal subgroup of | #72214572, #72214923, #72215603, … |
| maximal quotient of | C2×#72214173, #72240124, #72260936, … |
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, 3, F.7);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);