Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#35310 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3)(5,7)(8,9,13,12)(10,15,14,11)(16,17)(18,21)(19,22)(20,25)(23,28)(24,26)(27,30)(29,31);(8,10)(9,11)(12,15)(13,14)(17,18)(19,26)(23,25)(27,31);(4,5)(6,7)(8,11,13,15)(9,14,12,10)(16,18,21,17)(19,24,26,22)(20,25,28,23)(27,30,31,29);(8,12,13,9)(10,15,14,11)(16,19,21,26)(17,22,18,24)(20,27,28,31)(23,29,25,30);(16,20,21,28)(17,23,18,25)(19,27,26,31)(22,29,24,30)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28298);