Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C22 |
| direct product | C2×C11×#330 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C11 | C22⋊C4 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C42 | C23×C11 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C11 ◁ C2×C42×C11 ◁ G | |
232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1408 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29);(4,7,8,9)(11,12,15,17,16,13,14,18);(3,4)(5,9)(6,8)(7,10)(11,13,16,12)(14,17,15,18)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 255);