Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C4×#417 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
853 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4);(5,6,7,8)(10,13)(11,15)(14,16);(9,10)(11,16)(12,14)(13,15);(10,14)(13,16)88 conjugacy classes — view the full character table.
| maximal subgroup of | #105636, #105637, #106406, … |
| maximal normal subgroup of | #105636, #105637, #106406, … |
| maximal quotient of | C4×#1477, #182160, #182172, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13402);