Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of UT(4,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C35 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | C32⋊H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C33 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C33 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
223 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
Polycyclic presentation:
3^6|a^3=f,b^3=f,c^3=f,b^a=bd,c^a=ce,d^a=df,e^b=ef,d^c=df57 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 312);