Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C8×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C22 |
| commutator | C2 | C2×C4×C8 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C23 | C22×C4 |
| prime radical | C24 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(9,10,11,13,12,14,15,16);(1,2)(3,4)(5,6,7,8);(2,4)80 conjugacy classes — view the full character table.
| maximal subgroup of | #7199, #7202, C8×#99, … |
| maximal normal subgroup of | #7199, #7202, C8×#99, … |
| maximal quotient of | C8×C22⋊C8, #8145, #8148, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 483);