Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C23 |
| direct product | C3×#13264 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C8 | C23 |
| commutator | C23 | C22×C3×C8 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| socle, prime radical | C23×C3 | C22×C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C8 ◁ G | |
386 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)(16,17,21,26,22,24,29,31,20,25,27,30,19,18,23,28);(4,5,6,7)(9,12)(13,14)(16,18,27,31,22,17,23,30,20,24,21,28,19,25,29,26);(8,10,11,15)(9,13,12,14)(16,19,20,22)(17,24,25,18)(21,23,27,29)(26,31,30,28)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6252);