Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#7571 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C22⋊C4 |
| commutator | C2×C4 | C22×C3×C8 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle, prime radical | C23×C3 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C3×C43 ◁ G | |
516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(5,8)(6,10)(9,11)(13,17,18,19)(20,21,22,25,23,24,26,27);(4,5,7,9)(6,11,10,8)(12,13,16,18)(14,17,15,19)(21,24)(22,26);(12,14,16,15)(13,17,18,19)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 559);