Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | Aut(G) order 8388608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#2859 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1279 conjugacy classes of subgroups; only the 687 normal subgroups are shown.
This group has 1279 conjugacy classes of subgroups — too many to draw. Only its 687 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(4,6)(9,12)(10,14)(15,16,20,19)(17,22,21,18);(7,8,11,13)(9,10,12,14)(15,17)(16,18)(19,22)(20,21);(3,4,5,6)(7,9,11,12)(8,10,13,14)(15,18,20,22)(16,21,19,17);(7,10)(8,12)(9,13)(11,14)(15,19,20,16)(17,22,21,18)88 conjugacy classes — view the full character table.
| maximal subgroup of | #96929, #111331, #112528, … |
| maximal normal subgroup of | #96929, #111331, #112528, … |
| maximal quotient of | C2×#14687, C2×#14699, C2×#14707, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29616);