Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#472 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
244 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 100 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);(4,8)(5,12)(6,9)(11,16)(13,14)(15,17);(3,4,9,16,7,11,6,8)(5,10,12,17,14,18,13,15);(3,5)(4,10)(6,13)(7,14)(8,15)(9,12)(11,18)(16,17)32 conjugacy classes — view the full character table.
| maximal subgroup of | #12996, #13007, #13068, … |
| maximal normal subgroup of | #12996, #13007, #13068, … |
| maximal quotient of | C2×#1667, C2×#1675, C2×#1677, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1797);