Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#2102 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | #400 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
355 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 355 conjugacy classes of subgroups — too many to draw. Only its 151 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);(3,4,7,6,8,10,12,16)(5,11,13,18,14,15,17,9);(3,5,12,13,8,14,7,17)(4,9,16,11,10,18,6,15);(3,6,7,10,8,16,12,4)(5,11,13,18,14,15,17,9)52 conjugacy classes — view the full character table.
| maximal subgroup of | #137220, #137221, #137222, … |
| maximal normal subgroup of | #137220, #137221, #137222, … |
| maximal quotient of | C2×#10687, C2×#10689, C2×#10688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16738);