Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C4⋊C4×SD16 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
515 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,4)(5,6,7,8);(1,2,3,4);(9,10,12,15)(11,13,16,14);(10,13)(11,16)(14,15)70 conjugacy classes — view the full character table.
| maximal subgroup of | SD16×#102, #220803, SD16×#103, … |
| maximal normal subgroup of | SD16×#102, #220803, SD16×#103, … |
| maximal quotient of | SD16×C4⋊C8, #225457, #225473, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15669);