Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
| direct product | C2×#1380 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | D8 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
347 conjugacy classes of subgroups; only the 207 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 207 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(3,4)(5,6)(7,8,9,10)(12,17)(13,21)(14,23)(15,22)(18,19)(20,26)(24,25);(4,6)(11,12,14,18,15,19,13,17,16,20,21,25,22,24,23,26)88 conjugacy classes — view the full character table.
| maximal subgroup of | #94339, #94345, C4×#1380, … |
| maximal normal subgroup of | #94339, #94345, C4×#1380, … |
| maximal quotient of | C2×#7154, #126685, #126702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5543);