Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#34968 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| socle | C23×C3 | C25 |
| prime radical | C3×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
1940 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)(4,5)(6,11)(7,9)(8,10)(12,13)(14,19)(15,17)(16,18)(22,26)(24,27);(4,6,8,7)(5,9,10,11)(20,21)(22,27)(23,25)(24,26);(12,14,16,15)(13,17,18,19)(20,22,23,26)(21,24,25,27);(4,7,8,6)(5,9,10,11)(21,25)(24,27);(12,15,16,14)(13,17,18,19)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27956);