Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C33 · Schur multiplier (H2) = C29×C33 |
| direct product | C2×C33×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C33 | C24 |
| commutator, Frattini, cosocle | C2 | C25×C33 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C33 ◁ G | |
12572 conjugacy classes of subgroups; only the 10556 normal subgroups are shown.
This group has 12572 conjugacy classes of subgroups — too many to draw. Only its 10556 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2)(11,12,13);(3,4)(5,9)(6,7)(8,10)(14,15,16);(4,7)(8,9)(17,18,19);(3,4)(5,8)(6,7)(9,10);(3,5)(4,8)(6,10)(7,9)918 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47837);