Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×#356 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C22×Q8 | C23 |
| socle | C22 | C2×#99 |
| prime radical | C23 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
250 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 83 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4);(5,6,9,13,10,14,7,15)(8,12,16,19,18,20,17,11);(5,7,10,9)(6,11,14,19)(8,17,18,16)(12,13,20,15)52 conjugacy classes — view the full character table.
| maximal subgroup of | #94141, #94145, #94152, … |
| maximal normal subgroup of | #94141, #94145, #94152, … |
| maximal quotient of | C4×#1319, C4×#1325, #128369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5618);