Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) order 6291456 |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#2275 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator | C2 | C23×C42 |
| Frattini, cosocle | C23 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2347 conjugacy classes of subgroups; only the 1551 normal subgroups are shown.
This group has 2347 conjugacy classes of subgroups — too many to draw. Only its 1551 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,6,5)(7,8,10,9)(11,12,15,13)(14,17,18,16);(7,8,10,9)(12,13)(16,17);(3,5,6,4)(7,9,10,8)(11,13,15,12)(14,17,18,16);(11,14)(12,16)(13,17)(15,18)136 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2275⋊C3, #72268655 |
| maximal subgroup of | #105640, #105782, #105788, … |
| maximal normal subgroup of | #105640, #105782, #105788, … |
| maximal quotient of | C2×#13749, C2×#13750, C4×#2275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26985);