Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C22×D4 | C23 |
| socle | C22 | C2×#99 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
419 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 119 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 — 3 generators on 20 points:
(5,6)(7,11)(8,15,16,10)(9,12)(13,19,20,17)(14,18);(1,2,3,4)(6,10)(11,17)(12,15)(18,19);(5,7,9,14)(6,11,12,18)(8,13,16,20)(10,17,15,19)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=h,b^a=bd,d^a=dg,g^a=gh,e^b=eg,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #146352, #146353, #146484, … |
| maximal normal subgroup of | #146352, #146353, #146484, … |
| maximal quotient of | #126908, #126947, #126972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5561);