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×#35326 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3)(5,7)(9,12)(13,15)(16,17,21,25)(18,19,23,22)(20,28,29,24)(26,30,31,27);(4,5,6,7)(18,23)(19,22)(26,31)(27,30);(8,9,11,12)(10,13,14,15)(16,18,21,23)(17,22,25,19)(20,26,29,31)(24,30,28,27);(8,10,11,14)(9,13,12,15)(16,19,21,22)(17,23,25,18)(20,27,29,30)(24,31,28,26);(16,20,21,29)(17,24,25,28)(18,26,23,31)(19,27,22,30)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28314);