Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#2228 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
303 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12,13,16)(14,17,20,24)(15,23,21,19)(18,26,25,22);(3,5,6,10)(4,7,8,9)(12,16)(17,24)(18,25)(19,23);(12,17)(14,22)(15,21)(16,24)(18,19)(20,26)(23,25)58 conjugacy classes — view the full character table.
| maximal subgroup of | #138832, #138838, C4×#2228, … |
| maximal normal subgroup of | #138832, #138838, C4×#2228, … |
| maximal quotient of | C2×#12813, C2×#12935, C2×#12937, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26610);