Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C2×C5×#2196 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C5 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C5 ◁ C22×C5×D4 ◁ G | |
1956 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 1956 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2)(11,12,13,14,15);(3,4)(5,7)(6,8)(9,10);(4,7)(8,10);(5,9)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26531);