Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#551 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C24 |
| commutator, Frattini, cosocle | C22 | C25 |
| prime radical | C2×#120 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
664 conjugacy classes of subgroups; only the 404 normal subgroups are shown.
This group has 664 conjugacy classes of subgroups — too many to draw. Only its 404 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 — 5 generators on 18 points:
(1,2);(3,4,7,9)(5,6,10,8)(11,12,14,16)(13,15,18,17);(3,5,7,10)(4,8,9,6)(11,12)(13,17)(14,16)(15,18);(3,6)(4,5)(7,8)(9,10);(3,5)(4,6)(7,10)(8,9)(11,13)(12,15)(14,18)(16,17)50 conjugacy classes — view the full character table.
| maximal subgroup of | #12761, #17412, #17422, … |
| maximal normal subgroup of | #12761, #17412, #17422, … |
| maximal quotient of | C2×#2342, C2×#2343, C2×#2353, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2201);