Finite Group DatabasebetaFinite Group Databasebeta| order | 656 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C41 · Schur multiplier (H2) = C22 |
| direct product | C22⋊C4×C41 |
| unambiguous semidirect product | C22⋊(C4×C41) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C41 | C22 |
| commutator | C2 | C2×C4×C41 |
| Frattini, cosocle | C22 | C22×C41 |
| prime radical | C23×C41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C41 ◁ G | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 328 |
A minimal faithful permutation representation — 2 generators on 49 points:
(1,2,4,6)(3,5,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49);(2,5)(6,8)410 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(656, 21);