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×#34525 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2008 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)(4,5,7,6)(8,9)(10,11)(12,13)(14,15)(16,17,20,23)(18,24,26,21)(19,22,27,29)(25,30,31,28);(9,13)(10,14)(16,17,20,23)(18,21,26,24)(19,22,27,29)(25,28,31,30);(4,5,7,6)(16,17)(18,21)(19,22)(20,23)(24,26)(25,28)(27,29)(30,31);(8,10,12,14)(9,11,13,15)(16,18)(17,21)(19,25)(20,26)(22,28)(23,24)(27,31)(29,30);(4,6,7,5)(8,11)(9,14)(10,13)(12,15)(16,19,20,27)(17,22,23,29)(18,25,26,31)(21,28,24,30)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27513);