Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C9 · Schur multiplier (H2) = C35 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C32 | C34 |
| commutator | C3 | C33×C9 |
| prime radical | C3×H(3,𝔽3) | C32 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
463 conjugacy classes of subgroups.
| transitive degree | μt(G) = 243 |
Polycyclic presentation:
3^6|a^3=f,d^3=e,b^a=be,d^a=de,c^b=ce249 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 424);