Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#1361 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| prime radical | C2×#493 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
259 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 259 conjugacy classes of subgroups — too many to draw. Only its 87 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 — 3 generators on 26 points:
(1,2);(3,4,5,8,6,7,9,10)(11,12)(13,16)(14,23)(15,19)(17,20)(18,21)(22,25)(24,26);(4,7)(5,9)(12,16)(13,20)(14,21)(17,18)(19,25)(22,26)(23,24)58 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1361, #128900, #128933, … |
| maximal normal subgroup of | C4×#1361, #128900, #128933, … |
| maximal quotient of | C2×#7096, C2×#7097, #122872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4915);