Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#1700 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #121 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
325 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,7,8)(5,10,9,6)(11,12)(13,19)(14,16)(15,23)(17,21)(18,22)(20,26)(24,25);(3,5,6,4,7,9,10,8)(12,16)(17,25)(18,23)(19,26);(4,8)(5,9)(11,13,14,20)(12,17,16,25)(15,24,22,21)(18,26,23,19)46 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1700, #211439, #211459, … |
| maximal normal subgroup of | C4×#1700, #211439, #211459, … |
| maximal quotient of | C2×#8655, C2×#9226, C2×#9228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12140);