Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | M16×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | C24 |
| commutator, socle | C22 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(9,10,11,14,12,13,15,16);(10,13)(14,16);(1,2)(3,6)(4,5)(7,8);(2,5)(6,8);(1,3,4,7)(2,6,5,8)100 conjugacy classes — view the full character table.
| Fitting subgroup of | M16×#41⋊C3 |
| maximal subgroup of | #105658, #105663, #105944, … |
| maximal normal subgroup of | #105658, #105663, #105944, … |
| maximal quotient of | M16×#117, C4×D4×M16, C4×Q8×M16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53201);