Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C4×#543 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, socle, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1301 conjugacy classes of subgroups; only the 791 normal subgroups are shown.
This group has 1301 conjugacy classes of subgroups — too many to draw. Only its 791 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) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4);(5,6)(7,11)(8,10)(9,12)(13,14,16,18)(15,19,20,17);(5,7)(6,9)(8,12)(10,11);(7,12)(9,11)(13,15,16,20)(14,17,18,19);(6,10)(9,11)88 conjugacy classes — view the full character table.
| Fitting subgroup of | C4×#543⋊C3 |
| maximal subgroup of | #96952, #121985, #121987, … |
| maximal normal subgroup of | #96952, #121985, #121987, … |
| maximal quotient of | C4×#2329, C4×#2315, C4×#2437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27084);