Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C28 |
| direct product | C3×#7557 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25×C3 | C23 |
| commutator | C23 | C2×C3×C42 |
| Frattini, cosocle | C25 | C23×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25×C3 ◁ G | |
2886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(9,11)(12,13)(14,15)(17,18)(20,21,23,22);(5,7)(9,11)(13,15)(16,17,19,18)(20,22,23,21);(4,5)(6,7)(8,9,10,11)(12,13)(14,15)(16,18,19,17)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 545);