Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10624656 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C32 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C33 |
| prime radical | #6856 | C3 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C32×C9 ◁ G | |
| LECS | G ▷ C33 ▷ C32 ▷ C1 | |
99 conjugacy classes of subgroups.
| transitive degree | μt(G) = 243 |
Polycyclic presentation:
3^6|b^3=f^2,c^3=e,b^a=bd,c^a=cef,d^a=de,c^b=ce,d^b=df41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 237);