Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#3303 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×D4 ◁ G | |
398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7,9,11)(8,12,13,10)(14,15,19,22)(16,20,18,26)(17,24,25,28)(21,27,29,23);(6,8)(7,10)(9,13)(11,12)(14,16,19,18)(15,20,22,26)(17,23,25,27)(21,28,29,24);(14,17)(15,21)(16,23)(18,27)(19,25)(20,28)(22,29)(24,26);(8,13)(10,12)(14,16,19,18)(15,20,22,26)(17,23,25,27)(21,28,29,24)175 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18980);