Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#2165 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C22 | #432 |
| prime radical | C24 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
285 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 285 conjugacy classes of subgroups — too many to draw. Only its 159 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);(3,4,5,10,6,11,13,18,7,8,12,15,14,9,16,17);(4,8)(5,12)(9,11)(13,16);(4,9,8,11)(5,12)(10,18,15,17)(13,16)64 conjugacy classes — view the full character table.
| maximal subgroup of | #126894, #126895, #126896, … |
| maximal normal subgroup of | #126894, #126895, #126896, … |
| maximal quotient of | C2×#12595, C2×#12607, C2×#12659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26412);