Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) = #72303619 (order 4096) · Inn(G) = #99 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
| direct product | C2×#1321 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×M16 | C23 |
| socle | C22 | #323 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
259 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 259 conjugacy classes of subgroups — too many to draw. Only its 71 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);(3,4,7,12,8,11,22,28,9,13,23,30,6,18,20,29)(5,14,16,31,15,10,24,25,17,27,21,33,19,26,32,34);(3,5)(4,10)(6,19)(7,21)(8,15)(9,17)(11,27)(12,25)(13,26)(14,18)(16,23)(20,24)(22,32)(28,33)(29,31)(30,34)52 conjugacy classes — view the full character table.
| maximal subgroup of | #94092, #94093, #94094, … |
| maximal normal subgroup of | #94092, #94093, #94094, … |
| maximal quotient of | C2×#7056, C2×#7057, #122242, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4425);