Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#2134 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×Q8 | C24 |
| prime radical | C2×#535 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
695 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 695 conjugacy classes of subgroups — too many to draw. Only its 183 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);(3,4,7,12,8,13,5,10)(6,9,14,17,15,11,16,18);(4,9)(5,7)(6,15)(10,18)(11,13)(12,17);(3,5,8,7)(4,10,13,12)(6,14,15,16)(9,17,11,18)46 conjugacy classes — view the full character table.
| maximal subgroup of | #146651, #146671, #146698, … |
| maximal normal subgroup of | #146651, #146671, #146698, … |
| maximal quotient of | C2×#11433, C2×#11449, C2×#11479, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26355);