Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 6291456 |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C28×C4 |
| direct product | C22×C4×M16 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | C22 |
| commutator | C2 | C23×C42 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C24 | C22×C4 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1975 conjugacy classes of subgroups; only the 1631 normal subgroups are shown.
This group has 1975 conjugacy classes of subgroups — too many to draw. Only its 1631 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) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2);(3,4);(5,6,7,8);(9,10,11,12,13,14,15,16);(10,14)(12,16)160 conjugacy classes — view the full character table.
| Fitting subgroup of | C4×A4×M16 |
| maximal subgroup of | #105482, #105487, #105498, … |
| maximal normal subgroup of | #105482, #105487, #105498, … |
| maximal quotient of | C22×C8×M16, C22×#1451, C22×#1453, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53040);