Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C11 · Schur multiplier (H2) = C28 |
| direct product | C11×#3444 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C11 | C23 |
| commutator, Frattini, cosocle | C22 | C25×C11 |
| socle | C22×C11 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42×C11 ◁ G | |
1212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,17)(15,16)(18,19)(20,21)(22,25)(23,24)(26,27);(13,16)(17,19);(21,24)(25,27);(20,22,23,26)(21,25,24,27);(12,14,15,18)(13,17,16,19)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 2176);