Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C8 |
| direct product | C2×#1717 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #127 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×#493 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
483 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 483 conjugacy classes of subgroups — too many to draw. Only its 143 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) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(4,7)(5,9)(8,10)(12,17)(13,18)(14,15);(3,4,5,7,6,8,9,10)(11,12,15,13,16,18,14,17);(4,8)(7,10)(11,13,15,18,16,17,14,12)52 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1717, #213120, #213122, … |
| maximal normal subgroup of | C4×#1717, #213120, #213122, … |
| maximal quotient of | C2×#8821, C2×#9827, C2×#9829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12597);