Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #10624535 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C3×C9 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | #6879 |
| commutator, Frattini, cosocle | C92 | C32 |
| LCS | G ▷ C92 ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C32 ◁ C3×C9 ◁ C92 ◁ G | |
| LECS | G ▷ C92 ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 81 |
Polycyclic presentation:
3^6|b^3=d^2e,c^3=e^2f,d^3=f^2,b^a=bc,c^a=cd,d^a=de,e^a=ef89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 95);