Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C42×C8 · Schur multiplier (H2) = C23×C42 |
| direct product | C4×#336 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | C22 |
| commutator | C2 | C42×C8 |
| Frattini, cosocle | C23×C4 | C23 |
| socle, prime radical | C24 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
659 conjugacy classes of subgroups; only the 399 normal subgroups are shown.
This group has 659 conjugacy classes of subgroups — too many to draw. Only its 399 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) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4);(5,6)(7,8)(13,14,15,17,16,18,19,20);(6,8)(9,10,11,12)160 conjugacy classes — view the full character table.
| maximal subgroup of | #92806, #92814, #93288, … |
| maximal normal subgroup of | #92806, #92814, #93288, … |
| maximal quotient of | C4×#1273, #105246, #105247, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3680);