Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#421 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C4 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| socle | C22 | C2×C22⋊C4 |
| prime radical | C22×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
386 conjugacy classes of subgroups; only the 172 normal subgroups are shown.
This group has 386 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) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,7,10,8,11,6,15)(5,12,14,9,13,18,16,17);(3,5)(4,9)(6,16)(7,14)(8,13)(10,18)(11,17)(12,15);(5,13)(9,17)(12,18)(14,16)44 conjugacy classes — view the full character table.
| maximal subgroup of | #10772, #10773, #10775, … |
| maximal normal subgroup of | #10772, #10773, #10775, … |
| maximal quotient of | C2×#1478, C2×#1479, C2×#1477, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1632);