Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C22×#72232847 (order 8192) · Inn(G) = C23 · Out(G) = C23×#2846 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×#123 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C23 |
| commutator | C22 | C23×C4 |
| Frattini, socle, cosocle | C23 | C24 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
344 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16);(10,13)(14,16);(6,8)(9,11,12,15)(10,14,13,16)56 conjugacy classes — view the full character table.
| maximal subgroup of | #8966, #8968, #11472, … |
| maximal normal subgroup of | #8966, #8968, #11472, … |
| maximal quotient of | C4×#386, C4×#388, C4×#385, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1033);