Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#37266 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(5,8)(9,11)(12,13,16,14)(15,18,19,17)(20,21,25,28)(22,23,30,26)(24,32,33,27)(29,34,35,31);(13,14)(17,18)(20,22,25,30)(21,26,28,23)(24,29,33,35)(27,34,32,31);(4,5,7,8)(6,9,10,11)(12,14,16,13)(15,18,19,17);(12,15)(13,17)(14,18)(16,19)(20,23,25,26)(21,22,28,30)(24,31,33,34)(27,29,32,35);(4,6,7,10)(5,9,8,11)(20,24)(21,27)(22,29)(23,31)(25,33)(26,34)(28,32)(30,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30254);