Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×#333 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle, prime radical | C23 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
270 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 125 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);(5,6,7,10,8,9,11,12)(14,17)(15,19)(18,20);(6,9)(7,11)(13,14,16,18)(15,20,19,17)76 conjugacy classes — view the full character table.
| maximal subgroup of | #94133, #94370, C8×#333, … |
| maximal normal subgroup of | #94133, #94370, C8×#333, … |
| maximal quotient of | C4×#1272, #96742, #96746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1217);