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 |
| direct product | C2×#1391 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C2×#41 | C23 |
| socle | C22 | #342 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
391 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 391 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) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2);(3,4)(5,13)(6,17,18,10)(7,11)(8,12)(9,14,21,27)(15,25)(16,26)(19,30,31,23)(20,24)(22,28,32,34)(29,33);(3,5,7,15,8,16,20,29)(4,9,11,22,12,21,24,32)(6,14,19,28,18,27,31,34)(10,13,23,25,17,26,30,33)52 conjugacy classes — view the full character table.
| maximal subgroup of | #94283, #94285, #94287, … |
| maximal normal subgroup of | #94283, #94285, #94287, … |
| maximal quotient of | C2×#7188, #126949, #126975, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5564);