Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6875 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C27 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C9 | C9⋊C9 |
| commutator | C9 | C3×C27 |
| Frattini, cosocle | C3×C27 | C32 |
| socle | C3 | C9⋊C27 |
| prime radical | C32 | C27⋊C3 |
| LCS | G ▷ C9 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9 ◁ C3×C27 ◁ G | |
| LECS | G ▷ C3×C27 ▷ C9 ▷ C3 ▷ C1 | |
29 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 81 |
Polycyclic presentation:
3^6|a^3=d,b^3=c,c^3=f,d^3=e,e^3=f,b^a=bc,c^a=cf,d^b=df^2105 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10624532, #14546335, C3×#10624532, … |
| maximal normal subgroup of | C2×#10624532, #14546335, C3×#10624532, … |
| maximal quotient of | C2×#10624532, #14546334, #14546339, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 92);