Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
| direct product | C2×#1749 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C24 | C22×C4 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1143 conjugacy classes of subgroups; only the 447 normal subgroups are shown.
This group has 1143 conjugacy classes of subgroups — too many to draw. Only its 447 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);(3,4)(5,6)(7,8,10,9);(4,6)(11,12,13,16,14,15,17,18);(7,9,10,8)(12,15)(16,18)112 conjugacy classes — view the full character table.
| maximal subgroup of | #94944, #94948, #94952, … |
| maximal normal subgroup of | #94944, #94948, #94952, … |
| maximal quotient of | C2×#8145, C2×#8147, #181899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13361);