Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#1281 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22⋊C4 |
| commutator | C22 | C4×C5×C8 |
| Frattini, cosocle | C2×C42 | C22×C5 |
| socle | C22×C5 | #95 |
| prime radical | C23×C5 | C42 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3,4,5)(6,7,8,10,9,11,12,13)(14,15)(16,22,23,19)(17,21)(18,20)(24,28,29,26)(25,27);(14,16,17,24,18,23,25,29)(15,19,21,26,20,22,27,28)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 414);