Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#1683 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
363 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 363 conjugacy classes of subgroups — too many to draw. Only its 135 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)(5,7)(6,8)(9,10)(11,12,15,18)(13,19,22,26)(14,23,20,17)(16,24,25,21);(4,7)(8,9)(11,13,20,24,15,22,14,21)(12,16,23,26,18,25,17,19);(4,8)(6,10)(7,9)58 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1683, #211153, #211173, … |
| maximal normal subgroup of | C4×#1683, #211153, #211173, … |
| maximal quotient of | C2×#8648, C2×#9215, C2×#9219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12097);