Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#2166 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C22 | #432 |
| prime radical | C24 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
269 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 269 conjugacy classes of subgroups — too many to draw. Only its 159 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);(3,4,6,12,7,13,20,29,8,11,19,27,21,10,24,26)(5,14,16,28,17,25,31,34,15,9,18,23,30,22,32,33);(3,5)(4,9)(6,18)(7,17)(8,15)(10,25)(11,14)(12,28)(13,22)(16,19)(20,32)(21,30)(23,27)(24,31)(26,33)(29,34);(4,10,11,13)(5,15)(6,19)(9,22,14,25)(12,29,27,26)(17,30)(20,24)(23,34,28,33)64 conjugacy classes — view the full character table.
| maximal subgroup of | #126699, #126700, #126701, … |
| maximal normal subgroup of | #126699, #126700, #126701, … |
| maximal quotient of | C2×#12596, C2×#12608, C2×#12660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26413);