Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C11 · Schur multiplier (H2) = C28 |
| direct product | C2×C11×#563 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C11 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C11 |
| prime radical | C22×D4×C11 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C11 ◁ G | |
1024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37);(4,7)(5,9)(11,12,16,14)(13,19,21,17)(15,22,24,18)(20,25,23,26);(4,7)(8,10)(11,13,16,21)(12,17,14,19)(15,23,24,20)(18,26,22,25);(3,4)(5,8)(6,7)(9,10)(11,14,16,12)(13,17,21,19)(15,22,24,18)(20,25,23,26);(3,5)(4,8)(6,9)(7,10)(11,15)(12,18)(13,20)(14,22)(16,24)(17,25)(19,26)(21,23)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 2260);