Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C16 · Schur multiplier (H2) = C2 |
| direct product | C5×#1400 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | C22⋊C4 |
| commutator | C22 | C2×C5×C16 |
| Frattini, cosocle | C2×C16 | C22×C5 |
| socle | C2×C5 | C22⋊C16 |
| prime radical | C23×C5 | C16 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C2×C5×C16 ◁ G | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,3,4,5)(6,7,8,13,9,14,17,25,10,15,18,26,19,27,30,35,11,12,16,22,20,23,28,32,21,24,29,33,31,34,36,37);(7,12)(8,16)(14,23)(15,24)(17,28)(18,29)(27,34)(30,36)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 533);