Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#1830 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #381 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
609 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 609 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) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,7,12,8,11,21,26)(5,13,15,27,16,9,20,23)(6,17,19,31,18,10,24,25)(14,22,30,33,29,28,32,34);(3,5,8,16)(4,9,11,13)(6,14,18,29)(7,20,21,15)(10,22,17,28)(12,27,26,23)(19,32,24,30)(25,34,31,33);(3,6)(4,10,11,17)(5,14)(7,19)(8,18)(9,22,13,28)(12,25,26,31)(15,30)(16,29)(20,32)(21,24)(23,33,27,34)58 conjugacy classes — view the full character table.
| maximal subgroup of | #146501, #147607, #147608, … |
| maximal normal subgroup of | #146501, #147607, #147608, … |
| maximal quotient of | C2×#8735, #190015, C2×#8737, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13700);