Finite Group DatabasebetaFinite Group Databasebeta| order | 6561 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C37 · Schur multiplier (H2) = C320 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C9 | C36 |
| commutator, Frattini, cosocle | C3 | C37 |
| prime radical | H(7,𝔽3) | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9 ◁ G | |
| LECS | G ▷ C3 ▷ C1 | |
2057781 conjugacy classes of subgroups.
| transitive degree | μt(G) = 243 |
Polycyclic presentation:
3^8|a^3=h,c^b=ch,e^d=eh,g^f=gh2193 conjugacy classes. The character table is too large to display.
| maximal subgroup of | 3−1+8 |
| maximal normal subgroup of | 3−1+8 |
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");;
gap> coll := SingleCollector(F, [3, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);