Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#2177 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| prime radical | C22×D16 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
399 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 399 conjugacy classes of subgroups — too many to draw. Only its 167 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) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4,6,5)(7,8,10,13)(9,12,18,20)(11,19,17,15)(14,22,21,16);(8,12)(9,16)(11,17)(13,20)(14,15)(18,22)(19,21);(3,5,6,4)(8,13)(12,20)(14,21)(15,19)64 conjugacy classes — view the full character table.
| maximal subgroup of | #136703, #136707, #136737, … |
| maximal normal subgroup of | #136703, #136707, #136737, … |
| maximal quotient of | C2×#12045, C2×#12073, C2×#12086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26438);