Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×#1319 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #99 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C4⋊C4 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
309 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 309 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(3,4,5,7,6,8,9,10)(11,12)(13,19,21,26)(14,22,24,17)(15,18)(16,20)(23,25);(11,13,15,21)(12,16,18,25)(14,23,24,20)(17,26,22,19)52 conjugacy classes — view the full character table.
| maximal subgroup of | #94084, #94088, #94276, … |
| maximal normal subgroup of | #94084, #94088, #94276, … |
| maximal quotient of | C2×#7057, #122247, #122253, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4418);