Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#1569 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C23 | C2×C4⋊C4 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
379 conjugacy classes of subgroups; only the 215 normal subgroups are shown.
This group has 379 conjugacy classes of subgroups — too many to draw. Only its 215 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 — 4 generators on 18 points:
(1,2);(4,7)(5,9)(8,10)(11,12,13,16,14,15,17,18);(3,4,5,7,6,8,9,10);(4,8)(7,10)(12,15)(16,18)64 conjugacy classes — view the full character table.
| maximal subgroup of | #110004, #110005, #117829, … |
| maximal normal subgroup of | #110004, #110005, #117829, … |
| maximal quotient of | C2×#7876, C2×#8276, C4×#1569, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10782);