Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×C22⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C23 |
| commutator | C22 | C23×C4 |
| Frattini, socle, cosocle | C23 | C24 |
| prime radical | C24 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
328 conjugacy classes of subgroups; only the 172 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 172 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4);(5,6)(7,8)(9,10,12,14)(11,15,16,13);(9,11,12,16)(10,13,14,15);(6,8)56 conjugacy classes — view the full character table.
| maximal subgroup of | #7566, #7583, #7589, … |
| maximal normal subgroup of | #7566, #7583, #7589, … |
| maximal quotient of | C4×#385, C4×#387, C4×#389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1034);