Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#1304 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #95 |
| commutator | C23 | C42×C11 |
| Frattini, cosocle | C25 | C22×C11 |
| prime radical | C25×C11 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C25×C11 ◁ G | |
396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,20,22,25)(15,18)(16,19)(17,21,24,27)(23,26);(12,14)(13,17,19,25)(15,22)(16,21)(18,24,26,20)(23,27)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 36);