Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C11 · Schur multiplier (H2) = C28 |
| direct product | C11×#3483 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C11 |
| socle | C22×C11 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ G | |
1202 conjugacy classes of subgroups; only the 800 normal subgroups are shown.
This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 800 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 — 5 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,17)(18,19)(20,21)(22,25)(23,27)(24,26);(12,13,16,17)(14,18,15,19)(20,22,25,21)(23,27,24,26);(12,14,16,15)(13,18,17,19)(20,23,25,24)(21,26,22,27);(12,14,16,15)(13,19,17,18)(20,24,25,23)(21,26,22,27);(12,15,16,14)(13,19,17,18)550 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 2215);