Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#1344 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C23 |
| prime radical | C25 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
334 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(11,12)(13,19)(14,22,23,17)(15,18)(16,20,24,26)(21,25);(4,7)(5,10)(8,9)(11,13,15,21)(12,16,18,24)(14,20,23,26)(17,19,22,25)46 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1344, #128761, #128795, … |
| maximal normal subgroup of | C4×#1344, #128761, #128795, … |
| maximal quotient of | C2×#7077, C2×#7079, #122541, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4678);