Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C11 · Schur multiplier (H2) = C28 |
| direct product | C11×#3490 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C11 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ G | |
1328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,20)(15,19)(16,18)(17,21)(22,26)(23,25)(24,27);(12,14,16,22)(13,17,18,24)(15,21,23,27)(19,20,25,26);(13,18)(17,24)(19,25)(20,26);(14,21)(17,20)(22,27)(24,26);(13,19)(17,20)(18,25)(24,26)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 2222);