Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×AGL(1,ℤ/16) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #442 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#509 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
355 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 355 conjugacy classes of subgroups — too many to draw. Only its 151 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 — 4 generators on 18 points:
(1,2);(4,8,10,17)(5,12)(6,13)(9,15,18,11)(14,16);(3,4)(5,9)(6,15)(7,10)(8,16)(11,13)(12,17)(14,18);(3,5,13,16,7,14,6,12)(4,9,11,8,10,18,15,17)46 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26455);