Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×C3×#3131 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C23×C3 | C22×D4 |
| prime radical | C2×C3×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C22×C3×C42 ◁ G | |
2222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2)(27,28,29);(3,4)(5,8)(6,7)(9,10)(11,12)(13,19)(14,18)(15,20)(16,21)(17,22)(23,26)(24,25);(4,7)(8,10)(11,13,16,23,17,24,15,14)(12,18,20,25,22,26,21,19);(3,5,6,9)(4,8,7,10)(11,14,15,24,17,23,16,13)(12,19,21,26,22,25,20,18);(11,15,17,16)(12,20,22,21)(13,14,24,23)(18,25,26,19)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53593);