Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C8 |
| direct product | C2×#1709 |
| 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 | |
491 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 491 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) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(4,5)(6,7)(9,10)(11,12)(13,19)(14,22)(15,17)(16,20)(18,23)(21,25)(24,26);(3,4,7,9,8,10,6,5)(11,13,14,20,15,21,23,26)(12,16,18,19,17,24,22,25);(3,5,6,10,8,9,7,4)(12,17)(16,24)(18,22)(19,25)76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7092, #122859, #122865, … |
| maximal normal subgroup of | C2×#7092, #122859, #122865, … |
| maximal quotient of | C2×#8816, C2×#9822, C2×#9824, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12585);