Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#3291 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1341 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1341 conjugacy classes of subgroups — too many to draw. Only its 487 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 — 5 generators on 18 points:
(1,2);(3,4,9,14)(5,15,16,17)(6,11,10,18)(7,13,8,12);(3,5)(4,10)(6,14)(7,18)(8,11)(9,16)(12,17)(13,15);(3,6)(4,5)(7,17)(8,15)(9,10)(11,12)(13,18)(14,16);(3,6)(4,11)(5,12)(7,15)(8,17)(9,10)(13,16)(14,18)58 conjugacy classes — view the full character table.
| maximal subgroup of | #131836, #131841, #131843, … |
| maximal normal subgroup of | #131836, #131841, #131843, … |
| maximal quotient of | C2×#19884, C2×#19888, C2×#19890, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54090);