Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 2752512 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C23×#104 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C24 | C22⋊C4 |
| prime radical | C23×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
2879 conjugacy classes of subgroups; only the 1039 normal subgroups are shown.
This group has 2879 conjugacy classes of subgroups — too many to draw. Only its 1039 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 14 points:
(1,2);(3,4);(5,6);(7,8,10,13,11,9,14,12);(7,9)(8,11)(10,12)(13,14)112 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×A4×#104, #104×AGL(1,𝔽8) |
| maximal subgroup of | #123566, #123567, #123595, … |
| maximal normal subgroup of | #123566, #123567, #123595, … |
| maximal quotient of | C23×D4⋊C8, C23×Q8⋊C8, C22×#1476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53078);