Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#1516 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#525 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
749 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 749 conjugacy classes of subgroups — too many to draw. Only its 199 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) = 64 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4)(5,9,10,7)(6,8)(11,12)(13,19,21,26)(14,18)(15,17)(16,20,24,22)(23,25);(3,5)(4,7)(6,10)(8,9)(11,13)(12,16)(14,22)(15,21)(17,24)(18,26)(19,25)(20,23);(5,10)(7,9)(12,17)(16,24)(18,25)(19,26)64 conjugacy classes — view the full character table.
| maximal subgroup of | #105900, #107456, #107543, … |
| maximal normal subgroup of | #105900, #107456, #107543, … |
| maximal quotient of | C2×#7578, C2×#7579, C2×#7582, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10556);