Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#2161 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C22 | C2×#107 |
| prime radical | C24 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
321 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 321 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);(3,4,7,12,9,13,8,23)(5,14,18,30,16,28,19,33)(6,20,21,27,22,34,15,11)(10,24,26,32,25,31,29,17);(3,5,15,31,8,19,22,32,9,16,21,24,7,18,6,17)(4,10,20,30,12,26,27,28,13,25,34,33,23,29,11,14);(5,16)(10,25)(14,28)(17,32)(18,19)(24,31)(26,29)(30,33)64 conjugacy classes — view the full character table.
| maximal subgroup of | #147104, #147105, #147108, … |
| maximal normal subgroup of | #147104, #147105, #147108, … |
| maximal quotient of | C2×#12009, #483336, C2×#12025, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26401);