Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C11 · Schur multiplier (H2) = C25 |
| direct product | C11×#2931 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C11 | C23 |
| commutator | C22 | C23×C4×C11 |
| Frattini, cosocle | C2×C4 | C24×C11 |
| socle | C22×C11 | C23×C4 |
| prime radical | C11×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C11 ◁ G | |
500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,15)(16,17,21,24,20,23,29,31)(18,25,27,28,26,22,30,19);(13,15)(16,18)(17,22)(19,24)(20,26)(21,27)(23,25)(28,31)(29,30);(16,19,21,25,20,28,29,22)(17,18,24,27,23,26,31,30);(17,23)(18,26)(24,31)(27,30)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1663);