Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C4×#407 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C23 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C23×C4 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1123 conjugacy classes of subgroups; only the 407 normal subgroups are shown.
This group has 1123 conjugacy classes of subgroups — too many to draw. Only its 407 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4);(5,6)(7,8)(9,10,11,14,12,13,15,16);(6,8);(10,13)(14,16)112 conjugacy classes — view the full character table.
| maximal subgroup of | #94918, #94920, #94926, … |
| maximal normal subgroup of | #94918, #94920, #94926, … |
| maximal quotient of | C8×#407, #181920, #181921, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13364);