Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C33 · Schur multiplier (H2) = C25×C33 |
| direct product | C2×C33×#137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C33 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C24×C33 |
| prime radical | C2×C33×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C33 ◁ C22×C33×C4 ◁ G | |
3612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 4 generators on 27 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);(3,5,7,13)(4,8,9,17)(6,15,16,12)(10,18,14,11)(25,26,27);(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, 46092);