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×#3292 |
| 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 | |
1383 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1383 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)(5,16)(6,12)(7,14)(8,13)(9,15)(10,17)(11,18);(3,5,9,11)(4,10,15,6)(7,12,8,17)(13,18,14,16);(3,6)(4,11)(5,15)(7,18)(8,16)(9,10)(12,13)(14,17);(3,6)(4,12)(5,14)(7,16)(8,18)(9,10)(11,13)(15,17)58 conjugacy classes — view the full character table.
| maximal subgroup of | #131832, #131837, #131844, … |
| maximal normal subgroup of | #131832, #131837, #131844, … |
| maximal quotient of | C2×#19885, C2×#19893, C2×#19898, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54091);