Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 16777216 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#2310 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2607 conjugacy classes of subgroups; only the 1039 normal subgroups are shown.
This group has 2607 conjugacy classes of subgroups — too many to draw. Only its 1039 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 — 5 generators on 26 points:
(1,2);(4,6)(7,8,9,10)(12,16)(13,19)(14,22)(24,26);(11,12,15,16)(13,20,19,17)(14,18,22,23)(21,26,25,24);(3,4)(5,6)(7,8)(9,10)(11,13,15,19)(12,17,16,20)(14,21,22,25)(18,24,23,26);(11,14)(12,18)(13,21)(15,22)(16,23)(17,24)(19,25)(20,26)88 conjugacy classes — view the full character table.
| maximal subgroup of | #97019, #105819, #106515, … |
| maximal normal subgroup of | #97019, #105819, #106515, … |
| maximal quotient of | C2×#13776, C2×#13786, C4×#2310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27030);