Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 3870720 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C26×C3 |
| abelian | C24×C32×C5 |
| multiplicative | (ℤ/1736)ˣ, (ℤ/2232)ˣ, (ℤ/2604)ˣ, (ℤ/2772)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 720 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(9,10,11)(15,16,17,18,19);(3,4)(12,13,14);(5,6);(7,8)720 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 840);