Finite Group DatabasebetaFinite Group Databasebeta| order | 1568 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C49 · Schur multiplier (H2) = C25 |
| direct product | 2−1+4×C49 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C49 | C24 |
| commutator | C2 | C24×C49 |
| Frattini, socle, cosocle | C2×C7 | C24×C7 |
| prime radical | C7×2−1+4 | C7 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C49 ◁ G | |
219 conjugacy classes of subgroups.
| minimal degree | μ(G) = 65 |
| transitive degree | μt(G) = 784 |
A minimal faithful permutation representation — 4 generators on 65 points:
(2,6)(4,12)(7,9)(10,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65);(1,2,5,6)(3,9,11,7)(4,8,12,13)(10,15,16,14);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)833 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 197);