Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#42980 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2046 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)(8,9,12,14)(10,11,15,13)(16,17)(18,26)(19,28)(20,25)(21,22)(23,29)(24,30)(27,31);(8,10,12,15)(9,13,14,11)(17,22)(20,31)(23,26)(24,28);(4,5,6,7)(8,10)(9,11)(12,15)(13,14)(16,18,21,29)(17,23,22,26)(19,25,30,27)(20,28,31,24);(8,11)(9,10)(12,13)(14,15)(16,19,21,30)(17,24,22,28)(18,27,29,25)(20,23,31,26);(16,20,21,31)(17,25,22,27)(18,28,29,24)(19,23,30,26)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 35968);