Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 25165824 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C22×#398 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C23 |
| commutator, Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2079 conjugacy classes of subgroups; only the 903 normal subgroups are shown.
This group has 2079 conjugacy classes of subgroups — too many to draw. Only its 903 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(3,4);(1,2)(3,4);(9,10,11,12)(14,17)(18,19);(6,8)(13,14,16,17)(15,19,20,18);(5,6)(7,8)(9,10)(11,12)(13,15,16,20)(14,18,17,19)88 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#398 |
| maximal subgroup of | #97083, #97089, #111521, … |
| maximal normal subgroup of | #97083, #97089, #111521, … |
| maximal quotient of | C22×#1441, C22×#1442, C22×#1444, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27526);