Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#2178 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C22 | C4×D8 |
| prime radical | C2×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
331 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 331 conjugacy classes of subgroups — too many to draw. Only its 159 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 — 4 generators on 34 points:
(1,2);(3,4,8,12,9,14,24,13)(5,15,17,26,19,32,18,34)(6,11,20,29,21,30,7,22)(10,25,27,33,28,23,31,16);(3,5)(4,10)(6,16)(7,23)(8,17)(9,19)(11,26)(12,27)(13,31)(14,28)(15,22)(18,24)(20,25)(21,33)(29,32)(30,34);(3,6,8,20,9,21,24,7)(4,11,12,29,14,30,13,22)(5,16,17,25,19,33,18,23)(10,26,27,32,28,34,31,15)88 conjugacy classes — view the full character table.
| maximal subgroup of | #136677, #136678, #140731, … |
| maximal normal subgroup of | #136677, #136678, #140731, … |
| maximal quotient of | C2×C8×D16, C2×C8×SD32, C2×C8×Q32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26449);