Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#35758 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#2388 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2306 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);(8,9)(10,13)(11,12)(14,15)(17,21)(22,28)(23,29)(30,31);(4,5)(6,7)(8,9)(10,13)(11,12)(14,15)(16,17)(18,22)(19,23)(20,21)(24,30)(25,28)(26,31)(27,29);(8,10,11,14)(9,13,12,15)(16,18,20,25)(17,22,21,28)(19,26,27,24)(23,31,29,30);(16,19,20,27)(17,23,21,29)(18,24,25,26)(22,30,28,31)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28746);