Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#1985 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle, prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
719 conjugacy classes of subgroups; only the 303 normal subgroups are shown.
This group has 719 conjugacy classes of subgroups — too many to draw. Only its 303 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 — 4 generators on 22 points:
(1,2);(4,6)(7,8)(9,13)(10,11)(12,14)(15,16,20,21,19,22,18,17);(3,4,5,6)(8,11)(12,13)(15,17,18,22,19,21,20,16);(7,9,10,14)(8,12,11,13)(15,18,19,20)(16,21,22,17)88 conjugacy classes — view the full character table.
| maximal subgroup of | #106829, #106843, #107139, … |
| maximal normal subgroup of | #106829, #106843, #107139, … |
| maximal quotient of | C2×#9867, C2×#9868, C2×#9869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14287);