Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#1323 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C22⋊C4 |
| commutator | C2×C4 | C2×C8×C11 |
| Frattini, cosocle | C22×C8 | C22×C11 |
| socle | C22×C11 | C22⋊C8 |
| prime radical | C23×C11 | M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ C22×C8×C11 ◁ G | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 2 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,18,19,16)(15,17)(20,21,24,29,25,30,39,47,26,31,40,48,23,28,37,45)(22,32,34,49,33,43,50,46,35,44,38,51,36,27,41,42);(12,14,15,19)(13,16,17,18)(20,22)(21,27)(23,36)(24,38)(25,33)(26,35)(28,44)(29,46)(30,32)(31,43)(34,40)(37,50)(39,41)(42,48)(45,49)(47,51)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 55);