Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#1886 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
875 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 875 conjugacy classes of subgroups — too many to draw. Only its 247 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 — 4 generators on 22 points:
(1,2);(4,6)(7,8,13,16,12,17,11,15)(9,18,20,14,21,22,19,10);(3,4,5,6)(7,9)(8,14)(10,17)(11,20)(12,21)(13,19)(15,22)(16,18);(7,10,12,14)(8,9,17,21)(11,18,13,22)(15,20,16,19)64 conjugacy classes — view the full character table.
| maximal subgroup of | #146630, #146632, #146649, … |
| maximal normal subgroup of | #146630, #146632, #146649, … |
| maximal quotient of | C2×#8958, C2×#8962, #193808, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13938);