Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C4×#396 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C23 | C23×C4 |
| Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
983 conjugacy classes of subgroups; only the 295 normal subgroups are shown.
This group has 983 conjugacy classes of subgroups — too many to draw. Only its 295 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4);(5,6,7,8)(13,14)(15,16);(9,10)(11,12)(13,15,16,14);(6,8)(10,12)(14,15)88 conjugacy classes — view the full character table.
| Fitting subgroup of | C4×#396⋊C3 |
| maximal subgroup of | C8×#396, #359386, #359390, … |
| maximal normal subgroup of | C8×#396, #359386, #359390, … |
| maximal quotient of | C4×#1439, C4×#1441, C4×#1443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6853);