Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | M162 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C42 | C24 |
| commutator, socle | C22 | C22×C42 |
| prime radical | C24 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
450 conjugacy classes of subgroups; only the 307 normal subgroups are shown.
This group has 450 conjugacy classes of subgroups — too many to draw. Only its 307 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4,5,6,7,8);(2,6)(4,8);(9,10,11,12,13,14,15,16);(10,14)(12,16)100 conjugacy classes — view the full character table.
| maximal subgroup of | #108640, #108643, #108738, … |
| maximal normal subgroup of | #108640, #108643, #108738, … |
| maximal quotient of | M16×#97, #180837, #180855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10321);