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×#35246 |
| 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 | |
2294 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)(4,5,6,7)(9,12)(13,15)(16,17)(18,23)(19,22)(20,24)(21,26)(25,30)(27,29)(28,31);(4,5)(6,7)(16,18,22,21)(17,23,19,26)(20,28,29,25)(24,31,27,30);(8,9)(10,13)(11,12)(14,15)(16,19,22,17)(18,23,21,26)(20,27,29,24)(25,30,28,31);(16,20,22,29)(17,24,19,27)(18,25,21,28)(23,30,26,31);(8,10,11,14)(9,13,12,15)(16,21,22,18)(17,23,19,26)(20,28,29,25)(24,30,27,31)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28234);