Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C26 |
| direct product | C11×#1440 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C11 | C23 |
| commutator | C23 | C22×C4×C11 |
| Frattini, cosocle | C24 | C23×C11 |
| prime radical | C25×C11 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C11 ◁ G | |
736 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 736 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(17,19)(20,21,22,23)(24,25,27,26);(13,15)(17,19)(25,26);(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,26,27,25)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 172);