Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#36684 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| socle | C23×C3 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
2138 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)(8,9)(10,12)(11,13)(14,15)(16,17)(18,22)(19,23)(20,21)(24,27)(25,29)(26,28)(30,31);(8,10,11,15)(9,12,13,14)(17,21)(22,27)(23,28)(29,31);(5,7)(8,9)(10,14)(11,13)(12,15)(16,18)(17,22)(19,25)(20,24)(21,27)(23,29)(26,30)(28,31);(9,13)(12,14)(18,24)(22,27)(25,30)(29,31);(16,19,20,26)(17,23,21,28)(18,25,24,30)(22,29,27,31)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 29672);