Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×#352 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C23 | C2×C42 |
| Frattini, prime radical, cosocle | C22×D4 | C23 |
| socle | C22 | C2×#99 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
288 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4);(5,6)(7,13)(8,16,17,11)(9,12)(10,14,18,20)(15,19);(5,7,9,15)(6,10,12,18)(8,14,17,20)(11,13,16,19)52 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#352, #128451, #128470, … |
| maximal normal subgroup of | C8×#352, #128451, #128470, … |
| maximal quotient of | C4×#1317, C4×#1326, #128363, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5608);