Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C11 · Schur multiplier (H2) = C26 |
| direct product | C11×#2896 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C2×D4 |
| commutator | C4 | C23×C4×C11 |
| Frattini, cosocle | C2×C4 | C24×C11 |
| socle | C22×C11 | C22×D4 |
| prime radical | C22×C11×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ C23×C4×C11 ◁ G | |
772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 352 |
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)(18,25)(19,24)(20,28)(21,23)(22,26)(27,31)(29,30);(16,18)(17,22)(19,26)(20,29)(21,27)(23,30)(24,31)(25,28);(17,23)(22,30)(24,28)(25,31);(16,19,21,20)(17,24,23,28)(18,26,27,29)(22,31,30,25)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1628);