Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C42 |
| direct product | C2×#2375 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| prime radical | C22×#120 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2123 conjugacy classes of subgroups; only the 743 normal subgroups are shown.
This group has 2123 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) = 64 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(3,4,5,6)(8,11)(9,13)(15,16)(17,21)(18,19)(20,22);(8,11)(12,14)(16,19)(17,22);(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,22);(7,9)(8,12)(10,13)(11,14)(15,17)(16,20)(18,22)(19,21)76 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2375⋊C3 |
| maximal subgroup of | #146601, C2×#12960, C2×#13043, … |
| maximal normal subgroup of | #146601, C2×#12960, C2×#13043, … |
| maximal quotient of | C2×#13827, C2×#13831, C2×#13832, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27326);