Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C22⋊C4×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, socle, cosocle | C23 | C25 |
| prime radical | C23×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2823 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,4)(5,6,7,8);(2,4);(9,10)(11,14)(12,13)(15,16);(10,13)(14,16);(9,11,12,15)(10,14,13,16)100 conjugacy classes — view the full character table.
| Fitting subgroup of | C22⋊C4×#41⋊C3 |
| maximal subgroup of | #105934, #106850, #106972, … |
| maximal normal subgroup of | #105934, #106850, #106972, … |
| maximal quotient of | C22⋊C4×#117, #41×#95, #513494, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27090);