Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C23×C4×D4 (order 256) · Inn(G) = D4 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | D4⋊C16 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | D4 |
| commutator | C4 | C2×C16 |
| Frattini, cosocle | C4×C8 | C22 |
| socle | C22 | C22⋊C8 |
| prime radical | C2×D4 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
55 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24);(2,5)(3,8)(6,7)Polycyclic presentation:
2^7|a^2=d,c^2=e,d^2=f,f^2=g,b^a=bc,c^a=ce,c^b=ce56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D4⋊C16, #11582, #11583, … |
| maximal normal subgroup of | C2×D4⋊C16, #11582, #11583, … |
| maximal quotient of | #7068, #7069, #7070, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 61);