Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#2230 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #476 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
279 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 279 conjugacy classes of subgroups — too many to draw. Only its 119 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 — 4 generators on 18 points:
(1,2);(4,7)(5,13,15,10)(6,8)(11,12)(14,16,17,18);(3,4,6,7,9,12,8,11)(5,10,14,16,15,13,17,18);(3,5)(4,10)(6,14)(7,16)(8,17)(9,15)(11,18)(12,13)46 conjugacy classes — view the full character table.
| maximal subgroup of | #136857, #136859, #136861, … |
| maximal normal subgroup of | #136857, #136859, #136861, … |
| maximal quotient of | C2×#12823, C2×#12950, C2×#13116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26623);