Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#2022 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | C2×#534 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1197 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 1197 conjugacy classes of subgroups — too many to draw. Only its 191 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) = 64 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(7,8)(9,12)(10,13)(11,14)(15,18)(16,19)(17,21)(20,22);(7,9,11,16)(8,12,14,19)(10,15,17,22)(13,20,21,18);(4,6)(7,10)(8,13)(9,15)(11,17)(12,18)(14,21)(16,22)(19,20)52 conjugacy classes — view the full character table.
| maximal subgroup of | #137130, #137132, #137203, … |
| maximal normal subgroup of | #137130, #137132, #137203, … |
| maximal quotient of | C2×#9968, C2×#9969, C2×#9970, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16430);