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) = C24 |
| direct product | C2×#2090 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
405 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 405 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) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4)(5,10)(6,8)(7,9)(11,12,15,18,16,20,14,19)(13,17,21,26,23,24,22,25);(3,5,7,6)(4,8,9,10)(11,13)(12,17)(14,22)(15,21)(16,23)(18,26)(19,25)(20,24);(3,6,7,5)(4,8,9,10)(12,18)(13,21)(14,15)(17,24)(19,20)(22,23)52 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#2090, #359334, #377253, … |
| maximal normal subgroup of | C4×#2090, #359334, #377253, … |
| maximal quotient of | C2×#10653, C2×#10655, C2×#10656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16661);