Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = C22×D4×#120 (order 1024) · Inn(G) = C2×D4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C4 | C22×C8 |
| Frattini, cosocle | C42 | C23 |
| socle | C22 | C2×C22⋊C4 |
| prime radical | C2×#41 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
132 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,5)(3,7)(6,8)(9,10,11,13,12,14,15,16);(1,2)(3,8)(4,6)(5,7);(2,6)(5,8)Polycyclic presentation:
2^7|a^2=e,d^2=f,e^2=g,b^a=bd,d^a=df,c^b=cf,d^b=df44 conjugacy classes — view the full character table.
| maximal subgroup of | #8318, #8319, #8320, … |
| maximal normal subgroup of | #8318, #8319, #8320, … |
| maximal quotient of | #7574, #7577, #7578, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 209);