Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#416 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | D4 |
| commutator | C4 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| socle | C23 | C2×D4 |
| prime radical | C22×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
396 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 180 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4,5,6)(7,8)(9,16)(10,14)(11,19)(12,15)(13,17)(18,21)(20,22);(7,9)(8,13)(10,17)(11,20)(12,18)(14,21)(15,22)(16,19);(7,10,12,11)(8,14,15,19)(9,17,18,20)(13,21,22,16)56 conjugacy classes — view the full character table.
| maximal subgroup of | #10779, #10782, #10791, … |
| maximal normal subgroup of | #10779, #10782, #10791, … |
| maximal quotient of | C2×#1476, C2×#1488, #17443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1624);