Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) = C2×#1202 (order 2048) · Inn(G) = C24 · Out(G) = C2×D42 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | D4×M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C24 |
| commutator, socle | C22 | C23×C4 |
| prime radical | C22×D4 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
272 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2,3,4);(2,4);(5,6,7,8,9,10,11,12);(6,10)(8,12)50 conjugacy classes — view the full character table.
| maximal subgroup of | #8906, #8908, #8989, … |
| maximal normal subgroup of | #8906, #8908, #8989, … |
| maximal quotient of | M16×D8, #18035, M16×SD16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1666);