Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#9641 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C2×D4 |
| commutator | C2×C4 | C22×C3×C8 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle, prime radical | C23×C3 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C3×C43 ◁ G | |
530 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)(8,9,11,14,12,10,13,15)(16,17)(18,24)(19,27)(20,22)(21,26)(23,28)(25,30)(29,31);(8,10,11,15,12,9,13,14)(16,18,19,26,20,25,28,31)(17,21,23,24,22,29,27,30);(4,5,6,7)(17,22)(18,25)(23,27)(26,31)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2629);