Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#39360 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#547 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(9,11)(13,17)(14,20)(19,22)(21,26);(5,6)(12,13)(14,18)(15,22)(16,17)(19,23)(20,24)(21,27)(25,26);(4,5,7,6)(8,9,10,11);(12,14,16,20)(13,18,17,24)(15,21,23,26)(19,25,22,27);(4,6,7,5)(12,15,16,23)(13,19,17,22)(14,21,20,26)(18,25,24,27)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 32348);