Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10624589 |
| 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 | |
157 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 157 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 243 |
Polycyclic presentation:
3^6|b^3=f,c^3=e,b^a=bd,c^a=ce,d^a=de,d^b=df73 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10624613, C3×#10624613, #14553408, … |
| maximal normal subgroup of | C2×#10624613, C3×#10624613, #14553408, … |
| maximal quotient of | C2×#10624613, #14546639, #14546640, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 173);