Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#2382 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1099 conjugacy classes of subgroups; only the 743 normal subgroups are shown.
This group has 1099 conjugacy classes of subgroups — too many to draw. Only its 743 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);(3,4,5,6)(7,8)(9,13)(10,11)(12,14)(16,17)(20,21);(8,11)(9,14)(15,16,19,17)(18,21,22,20);(7,8)(9,12)(10,11)(13,14)(15,17,19,16)(18,21,22,20);(7,9,10,14)(8,12,11,13)(15,18,19,22)(16,20,17,21)76 conjugacy classes — view the full character table.
| maximal subgroup of | #107358, C2×#8912, C2×#8913, … |
| maximal normal subgroup of | #107358, C2×#8912, C2×#8913, … |
| maximal quotient of | C2×#13829, C2×#13833, C4×#2382, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27335);