Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C11 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×C11×#390 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C11 | C23 |
| commutator | C22 | C23×C4×C11 |
| Frattini, cosocle | C23 | C24×C11 |
| prime radical | C23×D4×C11 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C11 ◁ G | |
1360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(15,16,17,18,19,20,21,22,23,24,25);(4,6)(8,9)(11,12,14,13);(7,8,10,9)(11,13,14,12);(3,4,5,6)(7,9,10,8)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1024);