Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#1392 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C2×#41 | C23 |
| socle | C22 | #342 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
375 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 375 conjugacy classes of subgroups — too many to draw. Only its 119 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);(3,4,7,12,9,14,8,13)(5,15,17,30,19,32,18,31)(6,20,21,29,22,11,23,27)(10,24,25,16,26,33,28,34);(3,5,8,18,9,19,7,17)(4,10,13,25,14,26,12,28)(6,16,21,33,22,34,23,24)(11,15,27,31,20,32,29,30)52 conjugacy classes — view the full character table.
| maximal subgroup of | #94284, #94286, #94288, … |
| maximal normal subgroup of | #94284, #94286, #94288, … |
| maximal quotient of | C2×#7188, #126949, #126975, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5566);