Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#2238 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #481 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#469 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
302 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 302 conjugacy classes of subgroups — too many to draw. Only its 111 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 — 4 generators on 18 points:
(1,2);(4,8)(5,11)(6,14)(9,16)(10,13)(12,18)(15,17);(5,12)(8,16)(11,18)(13,17);(3,4,6,15,7,9,14,10)(5,13,11,16,12,17,18,8)40 conjugacy classes — view the full character table.
| maximal subgroup of | #137446, #137449, #137451, … |
| maximal normal subgroup of | #137446, #137449, #137451, … |
| maximal quotient of | C2×#12875, C2×#12931, C2×#13010, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26642);