Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C22×#349 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C22 | C23×C8 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C23 | C22⋊C8 |
| prime radical | C25 | C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
631 conjugacy classes of subgroups; only the 271 normal subgroups are shown.
This group has 631 conjugacy classes of subgroups — too many to draw. Only its 271 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 — 4 generators on 20 points:
(1,2);(3,4);(5,6,7,11,8,12,14,18,9,10,13,16,15,17,19,20);(6,10)(7,13)(12,17)(14,19)88 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#349 |
| maximal subgroup of | C2×#7191, #126920, C2×#7192, … |
| maximal normal subgroup of | C2×#7191, #126920, C2×#7192, … |
| maximal quotient of | C22×#1314, C2×#11418, C22×#1322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26323);