Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7076 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#7090 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #327 |
| commutator | C22×C8 | C2×C4×C5 |
| Frattini, cosocle | C2×C4⋊C8 | C22×C5 |
| prime radical | C23×C5 | #102 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×C4⋊C8 ◁ G | |
212 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(6,7,8,11,9,10,12,13)(15,19)(16,22)(17,24)(20,29)(21,28)(23,26)(25,27);(7,10)(14,15,18,21)(16,23,25,29)(17,26,24,20)(19,27,28,22)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 78);