Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#1292 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #95 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
345 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 345 conjugacy classes of subgroups — too many to draw. Only its 175 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 — 3 generators on 18 points:
(1,2);(4,8)(5,6)(9,10)(11,12,13,16,14,15,17,18);(3,4,6,9,7,10,5,8)(12,15)(13,17)64 conjugacy classes — view the full character table.
| maximal subgroup of | #93034, #93054, C4×#1292, … |
| maximal normal subgroup of | #93034, #93054, C4×#1292, … |
| maximal quotient of | C2×#7020, #95072, #95078, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 987);