Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#1681 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×#459 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
421 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 421 conjugacy classes of subgroups — too many to draw. Only its 135 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)(5,8)(9,10)(11,12)(13,15)(14,16)(17,18);(3,4,8,10,6,9,5,7)(12,15)(16,17);(12,16)(14,18)(15,17)58 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1681, #211152, #211172, … |
| maximal normal subgroup of | C4×#1681, #211152, #211172, … |
| maximal quotient of | C2×#8648, C2×#9217, C2×#9221, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12096);