Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×C3×#2383 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
2502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2)(23,24,25);(3,4,5,6)(8,11)(9,13)(15,16,19,17)(18,21,22,20);(8,11)(12,14)(16,17)(18,22);(7,8)(9,12)(10,11)(13,14)(15,17,19,16)(18,21,22,20);(7,9,10,13)(8,12,11,14)(15,18)(16,20)(17,21)(19,22)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27337);