Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 36864 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23×C3 |
| abelian | C22×C32×C4×C5 |
| multiplicative | (ℤ/1001)ˣ, (ℤ/1085)ˣ, (ℤ/1209)ˣ, (ℤ/1281)ˣ, (ℤ/1287)ˣ, (ℤ/1395)ˣ, (ℤ/1575)ˣ, (ℤ/1612)ˣ, (ℤ/1708)ˣ, (ℤ/2002)ˣ, (ℤ/2170)ˣ, (ℤ/2196)ˣ, (ℤ/2418)ˣ, (ℤ/2562)ˣ, (ℤ/2574)ˣ, (ℤ/2790)ˣ, (ℤ/3150)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2 | C23×C32×C5 |
| socle, prime radical | C23×C32×C5 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 720 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3,4)(9,10,11)(15,16,17,18,19);(5,6)(12,13,14);(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, 759);