Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×#10574511 (order 1024) · Inn(G) = D4 · Out(G) = C22×#131 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
| direct product | C2×#1401 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | D4 |
| commutator | C4 | C22×C16 |
| Frattini, cosocle | C2×C16 | C23 |
| socle | C22 | C22⋊C16 |
| prime radical | C2×#41 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
147 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 147 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 66 points:
(1,2);(3,4,7,13,8,14,26,39,9,15,27,40,29,42,54,62,10,16,28,41,30,43,55,63,6,12,23,36,24,37,51,60)(5,17,19,44,20,45,48,64,18,34,46,38,47,58,65,61,21,35,25,52,49,59,53,66,22,11,31,32,50,33,56,57);(3,5)(4,11)(6,22)(7,25)(8,20)(9,18)(10,21)(12,35)(13,38)(14,33)(15,17)(16,34)(19,28)(23,46)(24,50)(26,53)(27,31)(29,47)(30,49)(32,41)(36,44)(37,59)(39,61)(40,52)(42,45)(43,58)(48,55)(51,65)(54,56)(57,63)(60,64)(62,66)112 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1401, #151890, #151900, … |
| maximal normal subgroup of | C4×#1401, #151890, #151900, … |
| maximal quotient of | C2×#7335, #149522, C2×#7363, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6545);