Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | D4×C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C24 |
| commutator | C22 | C23×C4 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
388 conjugacy classes of subgroups; only the 196 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 196 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) = 64 |
A minimal faithful permutation representation — 4 generators on 12 points:
(10,12);(9,10)(11,12);(2,4)(5,6,7,8);(1,2,3,4)50 conjugacy classes — view the full character table.
| maximal subgroup of | #8882, #8886, #8890, … |
| maximal normal subgroup of | #8882, #8886, #8890, … |
| maximal quotient of | C22⋊C4×C4⋊C4, C4⋊C42, D4×#95, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1080);