Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#1394 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #99 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C22×Q8 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
409 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 409 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2);(7,8,10,14,11,15,19,22)(9,13,16,21,17,12,18,20);(3,4,5,6)(8,12,15,13)(14,21,22,20)52 conjugacy classes — view the full character table.
| maximal subgroup of | #94276, #94280, #94308, … |
| maximal normal subgroup of | #94276, #94280, #94308, … |
| maximal quotient of | C2×#7189, #126952, #126977, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5572);