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 | C2×C33×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C33×C4 | D4 |
| commutator, Frattini, cosocle | C4 | C24×C33 |
| socle | C22×C33 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C33×C4 ◁ C22×C33×C4 ◁ G | |
3724 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)(5,12)(6,10)(7,15)(8,11)(9,13)(14,17)(16,18)(22,23,24);(3,5)(4,9)(6,13)(7,16)(8,14)(10,17)(11,18)(12,15)(25,26,27);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,17,18,12)756 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46090);