Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C9 · Schur multiplier (H2) = C25×C3 |
| direct product | C2×C3×C9×#137 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C9 | C2×D4 |
| commutator | C4 | C24×C3×C9 |
| Frattini, cosocle | C3×C4 | C24×C32 |
| socle | C22×C32 | C2×C3×D4 |
| prime radical | C2×C32×#41 | C2×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C9 ◁ C22×C3×C4×C9 ◁ G | |
1290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2)(19,20,21);(3,4)(5,11)(6,14)(7,9)(8,12)(10,16)(13,18)(15,17)(22,23,24,25,26,27,28,29,30);(3,5,7,13)(4,8,9,17)(6,15,16,12)(10,18,14,11);(4,9)(8,17)(10,14)(11,18)594 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 20767);