Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C24 |
| direct product | C2×C11×#126 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C11 | C23 |
| commutator, Frattini, cosocle | C22 | C24×C11 |
| prime radical | C24×C11 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C11 ◁ G | |
246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29);(4,8)(5,12)(10,17)(11,15)(13,16)(14,18);(3,4,9,15)(5,10,16,17)(6,8,7,11)(12,18,13,14);(3,5,6,13)(4,10,8,14)(7,12,9,16)(11,18,15,17)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 1308);