Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | Aut(G) = #72216396 (order 1024) · Inn(G) = #386 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #386 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C24 | C23 |
| prime radical | #534 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
190 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 54 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)(3,6,10,14)(4,11,12,7)(5,8)(9,13)(15,16);(1,3)(2,6,8,14)(4,9)(5,10)(7,13,11,16)(12,15);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16)Polycyclic presentation:
2^7|a^2=f,b^2=df,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,e^b=eg,f^b=fg,d^c=dg,f^c=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | #12753, #12754, #12767, … |
| maximal normal subgroup of | #12753, #12754, #12767, … |
| maximal quotient of | #9107, #9108, #9109, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 645);