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×#34521 |
| 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)(9,12)(10,14)(16,17,22,20)(18,25,19,23)(21,28,29,24)(26,30,27,31);(9,12)(13,15)(16,18,22,19)(17,23,20,25)(21,26,29,27)(24,30,28,31);(4,5,7,6)(8,9,11,12)(10,13,14,15)(16,19,22,18)(17,23,20,25)(21,27,29,26)(24,30,28,31);(8,10)(9,13)(11,14)(12,15)(16,20,22,17)(18,23,19,25)(21,28,29,24)(26,30,27,31);(4,6,7,5)(16,21,22,29)(17,24,20,28)(18,26,19,27)(23,30,25,31)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27509);