Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C16 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#13263 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C8 | C23 |
| commutator | C22 | C22×C3×C16 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| socle, prime radical | C23×C3 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C8 ◁ G | |
398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3)(5,7)(8,9)(10,14)(11,12)(13,15)(16,17,19,23,20,24,27,30,21,18,22,25,28,26,29,31);(4,5,6,7)(9,12)(13,14)(16,18,19,25,20,26,27,31,21,17,22,23,28,24,29,30);(8,10,11,15)(9,13,12,14)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6251);