Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11992 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#11992 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1555 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| prime radical | C5×#3151 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C4×C5×M16 ◁ G | |
412 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 98 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 — 3 generators on 21 points:
(1,2,3,4,5)(6,7,8,14,9,13,17,21,10,12,16,20,18,11,15,19);(7,11)(8,15)(9,18)(12,13)(14,20)(16,17);(7,12)(8,16)(11,13)(15,17)140 conjugacy classes — view the full character table.
| Fitting subgroup of | AGL(1,𝔽5)×#11992 |
| maximal subgroup of | D5×#11992, C5×#489347 |
| maximal normal subgroup of | D5×#11992, C5×#489347 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4980);