Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C22×#477 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1179 conjugacy classes of subgroups; only the 607 normal subgroups are shown.
This group has 1179 conjugacy classes of subgroups — too many to draw. Only its 607 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:
(1,2);(3,4);(5,6,8,10)(7,11,12,9)(14,17)(15,18)(19,20);(5,7,8,12)(6,9,10,11)(13,14,16,17)(15,18,19,20);(13,15,16,19)(14,18,17,20)76 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#477 |
| maximal subgroup of | #111774, #111792, #111876, … |
| maximal normal subgroup of | #111774, #111792, #111876, … |
| maximal quotient of | C22×#1803, C22×#1863, C22×#1921, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53465);