Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | Aut(G) = #72225623 (order 1024) · Inn(G) = #99 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#352 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #99 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C2×D4 | C23 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
139 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 139 conjugacy classes of subgroups — too many to draw. Only its 44 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 — 3 generators on 18 points:
(1,2);(3,4)(5,11)(6,14,15,9)(7,10)(8,12,16,18)(13,17);(3,5,7,13)(4,8,10,16)(6,12,15,18)(9,11,14,17)26 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#352, #12621, #12688, … |
| maximal normal subgroup of | C4×#352, #12621, #12688, … |
| maximal quotient of | C2×#1317, C2×#1326, #11472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 851);