Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#35312 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2262 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)(5,6)(10,14)(12,15)(16,17,21,18)(19,22,26,24)(20,25,28,23)(27,30,31,29);(4,5,7,6)(8,9,11,13)(10,12,14,15)(17,18)(19,26)(23,25)(27,31);(4,6,7,5)(8,10)(9,12)(11,14)(13,15)(16,18,21,17)(19,24,26,22)(20,25,28,23)(27,30,31,29);(16,19)(17,22)(18,24)(20,27)(21,26)(23,29)(25,30)(28,31);(16,20,21,28)(17,23,18,25)(19,27,26,31)(22,29,24,30)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28300);