Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C22×D4×#120 (order 1024) · Inn(G) = C2×D4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C42 | C23 |
| socle | C22 | C2×C22⋊C4 |
| prime radical | C2×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
133 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 133 conjugacy classes of subgroups — too many to draw. Only its 52 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)(9,10,11,14,12,13,15,16);(1,2)(3,8)(4,6)(5,7);(2,6)(5,8)(10,13)(14,16)Polycyclic presentation:
2^7|a^2=e,d^2=f,e^2=g,b^a=bd,c^a=cg,d^a=df,c^b=cf,d^b=df38 conjugacy classes — view the full character table.
| maximal subgroup of | #8440, #8441, #8442, … |
| maximal normal subgroup of | #8440, #8441, #8442, … |
| maximal quotient of | #7612, #7613, #7614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 221);