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×#1692 |
| 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 135 normal subgroups are shown.
This group has 325 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) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(4,7)(5,9)(8,10)(11,12,15,18)(13,19,21,26)(14,22,24,17)(16,23,25,20);(3,4,5,7,6,8,9,10);(4,8)(7,10)(11,13,15,21)(12,16,18,25)(14,23,24,20)(17,26,22,19)46 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1692, #211427, #211447, … |
| maximal normal subgroup of | C4×#1692, #211427, #211447, … |
| maximal quotient of | C2×#8651, C2×#9224, C2×#9226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12135);