Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | Aut(G) = #72303611 (order 4096) · Inn(G) = #99 · Out(G) = C2×#386 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
| direct product | C2×#1328 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C23 | C22×C8 |
| Frattini, cosocle | C2×M16 | C23 |
| socle | C22 | #324 |
| prime radical | C24 | C2×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
189 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 189 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) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2);(3,4,6,11,7,12,17,25,8,10,16,22,18,23,29,32)(5,9,14,19,15,21,28,30,13,20,26,24,27,31,34,33);(3,5)(4,9,10,20)(6,14)(7,15)(8,13)(11,24,22,19)(12,21,23,31)(16,26)(17,28)(18,27)(25,33,32,30)(29,34)52 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1328, #128351, #128352, … |
| maximal normal subgroup of | C4×#1328, #128351, #128352, … |
| maximal quotient of | C2×#7064, #122313, #122316, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4445);