Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C33 · Schur multiplier (H2) = C25×C33 |
| direct product | C22×C33×Q16 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C33 | D4 |
| commutator, Frattini, cosocle | C4 | C24×C33 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C33 ◁ C22×C33×C4 ◁ G | |
3612 conjugacy classes of subgroups; only the 2492 normal subgroups are shown.
This group has 3612 conjugacy classes of subgroups — too many to draw. Only its 2492 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1728 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2)(21,22,23);(3,4)(24,25,26);(5,6,7,8,9,10,11,12)(13,20,19,18,17,16,15,14)(27,28,29);(5,13,9,17)(6,14,10,18)(7,15,11,19)(8,16,12,20)756 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46089);