Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C22×#471 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1379 conjugacy classes of subgroups; only the 527 normal subgroups are shown.
This group has 1379 conjugacy classes of subgroups — too many to draw. Only its 527 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(6,10)(7,13)(8,16)(12,18)(14,15)(17,19);(5,6,8,10,9,12,16,18)(7,11,14,17,15,20,13,19);(5,7)(6,11)(8,14)(9,15)(10,17)(12,20)(13,16)(18,19)64 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#471 |
| maximal subgroup of | #365017, #365020, #365045, … |
| maximal normal subgroup of | #365017, #365020, #365045, … |
| maximal quotient of | C22×#1665, C22×#1671, C22×#1681, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53451);