Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#3261 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
877 conjugacy classes of subgroups; only the 447 normal subgroups are shown.
This group has 877 conjugacy classes of subgroups — too many to draw. Only its 447 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(4,7)(5,9)(11,12,16,20)(13,17,22,31)(14,24,26,34)(15,27,29,19)(18,28,33,25)(21,35,36,42)(23,39,38,32)(30,40,41,37);(3,4)(5,8)(6,7)(9,10)(11,13,16,22)(12,17,20,31)(14,21,26,36)(15,23,29,38)(18,30,33,41)(19,32,27,39)(24,35,34,42)(25,37,28,40);(4,7)(8,10)(11,14,16,26)(12,18,20,33)(13,21,22,36)(15,28,29,25)(17,30,31,41)(19,34,27,24)(23,40,38,37)(32,42,39,35);(3,5,6,9)(4,8,7,10)(13,22)(17,31)(21,36)(23,38)(30,41)(32,39)(35,42)(37,40)52 conjugacy classes — view the full character table.
| maximal subgroup of | #131363, #131373, #131397, … |
| maximal normal subgroup of | #131363, #131373, #131397, … |
| maximal quotient of | C2×#21898, C2×#21968, C2×#22124, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53900);