Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C23×#137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C26 |
| prime radical | C23×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
5583 conjugacy classes of subgroups; only the 3183 normal subgroups are shown.
This group has 5583 conjugacy classes of subgroups — too many to draw. Only its 3183 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) = 128 |
A minimal faithful permutation representation — 6 generators on 22 points:
(1,2);(3,4);(5,6);(7,8)(9,15)(10,18)(11,13)(12,16)(14,20)(17,22)(19,21);(7,9,11,17)(8,12,13,21)(10,19,20,16)(14,22,18,15);(8,13)(12,21)(14,18)(15,22)88 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×A4×#137, #137×AGL(1,𝔽8) |
| maximal subgroup of | #357516, #357518, #357542, … |
| maximal normal subgroup of | #357516, #357518, #357542, … |
| maximal quotient of | C23×#417, C23×#419, C23×#428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56070);