Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) = #72234751 (order 4096) · Inn(G) = C2×C22⋊C4 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C25 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3,8,5,7,10,14)(4,6,9,13,11,12,15,16);(2,6)(3,9)(7,12)(10,15);(2,7)(6,12)(8,14)(13,16)Polycyclic presentation:
2^7|a^2=e,e^2=g,b^a=bd,c^a=cg,d^a=df,e^b=ef32 conjugacy classes — view the full character table.
| maximal subgroup of | #7213, #7324, #8346, … |
| maximal normal subgroup of | #7213, #7324, #8346, … |
| maximal quotient of | #7606, #7609, #7612, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 194);