Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#35885 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1750 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)(6,7)(8,10)(9,11)(14,18)(17,19)(21,25)(22,27);(4,6)(5,8)(7,11)(9,10)(12,13,16,15)(14,17,18,19)(20,21)(22,26)(23,27)(24,25);(4,7,9,8)(5,10,11,6)(12,14)(13,17)(15,19)(16,18)(20,22,24,27)(21,26,25,23);(4,6)(5,7)(8,11)(9,10)(12,15,16,13)(14,19,18,17)(20,23,24,26)(21,27,25,22);(4,8,9,7)(5,10,11,6)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28873);