Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = C2×#72217166 (order 4096) · Inn(G) = D16 · Out(G) = C22×#131 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#1417 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D16 |
| commutator | C16 | C22×C4 |
| Frattini, cosocle | C2×C16 | C23 |
| socle | C22 | #357 |
| prime radical | C2×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
147 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 147 conjugacy classes of subgroups — too many to draw. Only its 63 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);(3,4,7,12,9,15,28,14)(5,16,18,45,21,46,20,52)(6,22,25,53,27,55,60,54)(8,29,24,13,31,41,59,43)(10,32,33,48,35,47,44,17)(11,36,38,61,40,63,39,26)(19,50,49,64,51,65,30,37)(23,42,56,62,58,66,57,34);(3,5)(4,10)(6,23)(7,18)(8,30)(9,21)(11,37)(12,33)(13,42)(14,44)(15,35)(16,22)(17,26)(19,24)(20,28)(25,56)(27,58)(29,34)(31,49)(32,36)(38,50)(39,65)(40,64)(41,62)(43,66)(45,53)(46,55)(47,63)(48,61)(51,59)(52,54)(57,60)76 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1417, #151962, #151963, … |
| maximal normal subgroup of | C4×#1417, #151962, #151963, … |
| maximal quotient of | C2×#7383, C2×#7384, #150056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6607);