Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C28 |
| direct product | C3×#34488 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C25 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1940 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3)(4,5,7,6)(8,9)(10,14)(11,13)(12,15)(16,17)(18,21)(19,24)(20,25)(22,26)(23,28)(27,29)(30,31);(8,9,11,13)(10,12,15,14)(17,18)(19,26)(23,25)(27,31);(16,18,21,17)(19,24,26,22)(20,25,28,23)(27,30,31,29);(4,6,7,5)(8,9)(10,12)(11,13)(14,15)(16,19)(17,22)(18,24)(20,27)(21,26)(23,29)(25,30)(28,31);(8,10,11,15)(9,12,13,14)(16,20,21,28)(17,23,18,25)(19,27,26,31)(22,29,24,30)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27476);