Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#2046 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
729 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 729 conjugacy classes of subgroups — too many to draw. Only its 175 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)(7,8,11,15)(9,16,17,13)(10,12,19,14)(18,21,22,20);(8,12)(9,17)(10,19)(13,21)(14,15)(16,20);(4,6)(7,9,10,18,11,17,19,22)(8,13,14,21,15,16,12,20)52 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#2046, #359097, #376237, … |
| maximal normal subgroup of | C4×#2046, #359097, #376237, … |
| maximal quotient of | C2×#10514, C2×#10515, #238810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16506);