Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#35318 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3)(5,7)(9,13)(10,15)(16,17,21,19)(18,22,24,25)(20,27,28,23)(26,31,30,29);(8,9)(10,14)(11,15)(12,13)(17,19)(18,24)(23,27)(26,30);(4,5,6,7)(8,10,12,15)(9,14,13,11)(16,18)(17,22)(19,25)(20,26)(21,24)(23,29)(27,31)(28,30);(8,11,12,14)(9,15,13,10)(16,19,21,17)(18,25,24,22)(20,27,28,23)(26,31,30,29);(16,20,21,28)(17,23,19,27)(18,26,24,30)(22,29,25,31)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28306);