Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#1828 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #381 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
777 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 777 conjugacy classes of subgroups — too many to draw. Only its 223 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,11,13,8)(6,14)(7,10)(9,15)(12,16,18,17);(3,5)(4,8)(6,12)(7,13)(9,16)(10,11)(14,17)(15,18);(3,6)(4,9,10,14)(5,12)(7,15)(8,16,11,17)(13,18)58 conjugacy classes — view the full character table.
| maximal subgroup of | #146497, #146500, #147603, … |
| maximal normal subgroup of | #146497, #146500, #147603, … |
| maximal quotient of | C2×#8734, C2×#8735, #190013, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13697);