Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | Aut(G) = #72303619 (order 4096) · Inn(G) = #99 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
| direct product | C2×#1320 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C23 | C22×C8 |
| Frattini, cosocle | C2×M16 | C23 |
| socle | C22 | #323 |
| prime radical | C25 | C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
313 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2);(3,4,5,9,6,10,12,16,7,8,11,14,13,15,17,18);(4,8)(10,15)52 conjugacy classes — view the full character table.
| maximal subgroup of | #94076, #94077, #94078, … |
| maximal normal subgroup of | #94076, #94077, #94078, … |
| maximal quotient of | C2×#7054, C2×#7055, #122224, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4423);