Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C2×C3×#2634 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
2286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 5 generators on 33 points:
(1,2)(31,32,33);(8,11)(12,14)(15,16,20,18)(17,23,25,21)(19,22,27,26)(24,29,30,28);(4,6)(8,11)(12,14)(15,17,20,25)(16,21,18,23)(19,24,27,30)(22,28,26,29);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,18,20,16)(17,21,25,23)(19,22,27,26)(24,29,30,28);(7,9,10,13)(8,12,11,14)(15,19)(16,22)(17,24)(18,26)(20,27)(21,28)(23,29)(25,30)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27588);