Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
| direct product | C22×D42 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
10019 conjugacy classes of subgroups; only the 3479 normal subgroups are shown.
This group has 10019 conjugacy classes of subgroups — too many to draw. Only its 3479 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 12 points:
(1,2);(3,4);(5,6,7,8);(6,8);(9,10,11,12);(10,12)100 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55683);