Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | Aut(G) = C2×#60399 (order 512) · Inn(G) = #99 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #99 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C2×D4 | C23 |
| socle | C2 | C2×#99 |
| prime radical | #525 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
132 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,7)(4,11,12,6)(5,8)(9,15,16,13)(10,14);(2,6)(7,13)(8,11)(14,15);(1,3,5,10)(2,7,8,14)(4,9,12,16)(6,13,11,15)Polycyclic presentation:
2^7|a^2=e,c^2=g,b^a=bd,d^a=df,f^a=fg,e^b=ef,e^d=eg26 conjugacy classes — view the full character table.
| maximal subgroup of | #12623, #12624, #12711, … |
| maximal normal subgroup of | #12623, #12624, #12711, … |
| maximal quotient of | #11425, #11452, #11471, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 852);