Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7527 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C8 |
| direct product | C5×#7527 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1408 |
| commutator | C4×C8 | C2×C4×C5 |
| Frattini, cosocle | #492 | C22×C5 |
| prime radical | C5×#1721 | C2 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C5 ◁ C2×C4×C5 ◁ C42×C5 ◁ C5×#492 ◁ G | |
164 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,4,5)(6,7)(8,15,18,11)(9,20,17,14)(10,13)(12,16,21,19);(7,11)(8,16)(9,17)(12,20)(13,21)(14,15)(18,19)95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 515);