Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#2022 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #120 |
| commutator, Frattini, cosocle | C24 | C23×C5 |
| prime radical | C5×#534 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C24×C5 ◁ G | |
430 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,15)(13,16)(14,17)(18,21)(19,22)(20,24)(23,25);(10,12,14,19)(11,15,17,22)(13,18,20,25)(16,23,24,21);(7,9)(10,13)(11,16)(12,18)(14,20)(15,21)(17,24)(19,25)(22,23)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5612);