Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
| direct product | C4×#382 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C22 | C22×C42 |
| Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
871 conjugacy classes of subgroups; only the 447 normal subgroups are shown.
This group has 871 conjugacy classes of subgroups — too many to draw. Only its 447 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 — 4 generators on 20 points:
(1,2,3,4);(6,7)(9,10,11,12)(13,14)(15,18)(16,17)(19,20);(5,6,8,7)(14,17)(18,20);(5,7,8,6)(13,15,16,19)(14,18,17,20)112 conjugacy classes — view the full character table.
| maximal subgroup of | #94979, #94980, #94991, … |
| maximal normal subgroup of | #94979, #94980, #94991, … |
| maximal quotient of | C4×C4⋊#95, C4×#1434, C4×#1442, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6761);