Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C8×#102 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | D4 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C23 | C4×D4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
303 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,7,9,8,10,11,12);(1,2,3,4)(13,14)(15,17)(16,19)(18,20);(14,17)(15,20)(18,19)112 conjugacy classes — view the full character table.
| maximal subgroup of | C16×#102, #128659, #128660, … |
| maximal normal subgroup of | C16×#102, #128659, #128660, … |
| maximal quotient of | C8×D4⋊C8, #96694, #96709, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1186);