Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C25 |
| direct product | C11×#3277 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C11 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C4×D4×C11 ◁ G | |
476 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)(17,21)(19,27)(22,28)(23,30)(24,31)(25,29);(16,17,20,22)(18,24,26,30)(19,28,27,21)(23,29,31,25);(13,15)(17,22)(21,28)(23,31)(24,30);(16,18,20,26)(17,23,22,31)(19,29,27,25)(21,30,28,24)319 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 2009);