Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#2253 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C11 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C11 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C42×C11 ◁ C4×C8×C11 ◁ G | |
116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 1408 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,15,17)(14,18,19,16)(20,21,23,26)(22,30,32,44)(24,37,35,50)(25,38,36,29)(27,41,39,33)(28,43,40,31)(34,49,48,46)(42,47,45,51);(12,14,15,19)(13,16,17,18)(21,26)(27,39)(28,40)(29,38)(30,44)(37,50)(42,45)(46,49);(20,22,31,47,25,34,35,41,23,32,43,51,36,48,24,33)(21,27,37,46,29,42,40,44,26,39,50,49,38,45,28,30)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 985);