Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C22×C3×C42 (order 192) · Inn(G) = C1 · Out(G) = C22×C3×C42 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C720 |
| abelian | C5×C9×C16 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3×C8 | C2×C3×C5 |
| socle, prime radical | C2×C3×C5 | C3×C8 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 720 |
A minimal faithful permutation representation — 1 generator on 30 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25)(26,27,28,29,30)720 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 4);