Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#2660 |
| 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 | |
1231 conjugacy classes of subgroups; only the 607 normal subgroups are shown.
This group has 1231 conjugacy classes of subgroups — too many to draw. Only its 607 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);(7,8)(9,11)(10,12)(13,14)(15,16,19,17)(18,21,22,20);(3,4,5,6)(7,8,10,12)(9,11,14,13)(15,17,19,16)(18,20,22,21);(3,4)(5,6)(7,8)(9,13)(10,12)(11,14)(15,18,19,22)(16,20,17,21);(7,9,10,14)(8,11,12,13)64 conjugacy classes — view the full character table.
| maximal subgroup of | #111923, #111928, #112279, … |
| maximal normal subgroup of | #111923, #111928, #112279, … |
| maximal quotient of | C2×#14276, C2×#14277, C2×#14282, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27611);