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×#35319 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C24×C3 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2026 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)(8,9,11,13)(10,12,14,15)(17,22)(20,28)(23,24)(26,27);(8,9)(10,12)(11,13)(14,15)(16,17)(18,24)(19,23)(20,25)(21,22)(26,31)(27,30)(28,29);(4,5,6,7)(8,9)(10,12)(11,13)(14,15)(16,18,21,19)(17,23,22,24)(20,26,28,27)(25,30,29,31);(8,10,11,14)(9,12,13,15)(16,19,21,18)(17,24,22,23)(20,27,28,26)(25,31,29,30);(16,20,21,28)(17,25,22,29)(18,26,19,27)(23,30,24,31)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28307);