Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C22×#413 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C23×#41 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
2559 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(5,6,9,14,10,13,8,12)(7,15,16,20,17,11,18,19);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20);(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)88 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#413 |
| maximal subgroup of | #123517, #123519, #123523, … |
| maximal normal subgroup of | #123517, #123519, #123523, … |
| maximal quotient of | C22×#1458, C22×#1459, C2×#17349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53061);