Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = #72234665 (order 2048) · Inn(G) = C22⋊C4 · Out(G) = C22×H(5,𝔽2) |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×#99 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22⋊C4 |
| commutator | C22 | C2×C42 |
| Frattini, cosocle | C24 | C23 |
| socle | C22 | C2×C22⋊C4 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
206 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 206 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 3 generators on 12 points:
(1,2,3,4);(5,6)(7,11,12,9)(8,10);(5,7)(6,9)(8,12)(10,11)44 conjugacy classes — view the full character table.
| maximal subgroup of | #7204, #7205, #7282, … |
| maximal normal subgroup of | #7204, #7205, #7282, … |
| maximal quotient of | C4×#323, C4×#324, #8152, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 486);