Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#1275 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C42 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C23 | #95 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
339 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 339 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(3,4,5,7,6,8,9,10)(11,12)(13,20,22,26)(14,18)(15,24)(16,19)(17,21,25,23);(11,13,15,23,16,22,14,21)(12,17,18,20,19,25,24,26)88 conjugacy classes — view the full character table.
| maximal subgroup of | #93394, #93395, C4×#1275, … |
| maximal normal subgroup of | #93394, #93395, C4×#1275, … |
| maximal quotient of | C2×#7014, #94863, #94866, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 847);