Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C9 · Schur multiplier (H2) = C1 |
| direct product | C5×C9×Q16 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C9 | D4 |
| commutator | C4 | C22×C5×C9 |
| Frattini, cosocle | C3×C4 | C22×C3×C5 |
| socle, prime radical | C2×C3×C5 | C3×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C9 ◁ C4×C5×C9 ◁ G | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 720 |
A minimal faithful permutation representation — 2 generators 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)(23,30,29,28,27,26,25,24);(15,23,19,27)(16,24,20,28)(17,25,21,29)(18,26,22,30)315 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 88);