Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×#500 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle, prime radical | C23 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
439 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 439 conjugacy classes of subgroups — too many to draw. Only its 231 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 — 4 generators on 20 points:
(1,2,3,4);(5,6,8,10)(7,11,12,9)(14,17)(15,18)(19,20);(5,7,8,12)(6,9,10,11);(13,14,18,20,16,19,15,17)88 conjugacy classes — view the full character table.
| maximal subgroup of | #99596, #99606, #101992, … |
| maximal normal subgroup of | #99596, #99606, #101992, … |
| maximal quotient of | C8×#500, #213371, #213381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14708);