Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#2582 |
| 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 | |
1351 conjugacy classes of subgroups; only the 647 normal subgroups are shown.
This group has 1351 conjugacy classes of subgroups — too many to draw. Only its 647 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)(7,8,11,9)(10,13,14,12)(15,16,19,22)(17,21,18,20);(8,9)(12,13)(15,17,19,18)(16,20,22,21);(3,4)(5,6)(7,9,11,8)(10,13,14,12)(15,16)(17,20)(18,21)(19,22);(7,10,11,14)(8,12,9,13)(15,18,19,17)(16,21,22,20)76 conjugacy classes — view the full character table.
| maximal subgroup of | #97436, #111524, #115626, … |
| maximal normal subgroup of | #97436, #111524, #115626, … |
| maximal quotient of | C2×#13926, C2×#14017, C2×#14041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27937);