Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#1359 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #331 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| prime radical | C2×#127 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
221 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 221 conjugacy classes of subgroups — too many to draw. Only its 71 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 — 3 generators on 18 points:
(1,2);(3,4,5,10,13,8,12,16,7,11,15,18,6,9,14,17);(4,8)(5,12)(6,13)(9,11)(10,18)(14,15)46 conjugacy classes — view the full character table.
| maximal subgroup of | #94775, #94781, C4×#1359, … |
| maximal normal subgroup of | #94775, #94781, C4×#1359, … |
| maximal quotient of | C2×#7094, C2×#7095, #122897, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4909);