Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1402 |
| automorphism | Aut(G) = #72249492 (order 2048) · Inn(G) = AΣL(1,𝔽16) · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#1402 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | AΣL(1,𝔽16) |
| commutator | C2×D4 | C22×C4 |
| Frattini, cosocle | #127 | C23 |
| prime radical | C2×#453 | C2 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
282 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2);(3,4)(5,12,13,8)(6,10,16,9)(7,11)(14,18,15,17);(3,5)(4,8)(6,15)(7,14)(9,12)(10,18)(11,17)(13,16)34 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1402, #151906, #152009, … |
| maximal normal subgroup of | C4×#1402, #151906, #152009, … |
| maximal quotient of | C2×#7344, C2×#7375, #149673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6547);