Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C25×D4 |
| commutator, Frattini, cosocle | C2×C4 | C27 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#3449 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
110129 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^10|h^2=j,c^b=ch,d^c=dj,f^a=fj,g^a=gi,g^e=gj,h^b=hj,h^c=hj202 conjugacy classes — view the full character table.
| maximal subgroup of | #72251159, #72256305, #72262438, … |
| maximal normal subgroup of | #72251159, #72256305, #72262438, … |
| maximal quotient of | #72256907, #72264089, #72308100, … |
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, 8, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);