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 | H(5,𝔽2)×C49 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C49 | C24 |
| commutator | C2 | C24×C49 |
| Frattini, socle, cosocle | C2×C7 | C24×C7 |
| prime radical | C7×H(5,𝔽2) | C7 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C49 ◁ G | |
249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 392 |
A minimal faithful permutation representation — 4 generators on 57 points:
(1,2)(3,7)(4,5)(6,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57);(2,5)(6,7);(1,2)(3,6)(4,5)(7,8);(1,3)(2,6)(4,8)(5,7)833 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 196);