Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C22×C3×#400 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25×C3 | C23 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25×C3 ◁ G | |
3550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2)(25,26,27);(3,4);(11,15)(13,16)(17,18,20,22)(19,23,24,21);(6,8)(9,10,12,14)(11,16,15,13)(17,18)(19,23)(20,22)(21,24);(5,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,20,24)(18,21,22,23)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27529);