Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1412 |
| automorphism | Aut(G) = #72318502 (order 4096) · Inn(G) = #354 · Out(G) = #534 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#1412 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #354 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | #127 | C23 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
195 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 195 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)(5,12,14,18)(6,10,16,9)(7,11)(8,13,17,15);(3,5,7,13)(4,8)(6,15,16,14)(9,12)(10,17)(11,18)28 conjugacy classes — view the full character table.
| maximal subgroup of | #94783, C4×#1412, #151934, … |
| maximal normal subgroup of | #94783, C4×#1412, #151934, … |
| maximal quotient of | C2×#7358, C2×#7367, #149701, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6581);