Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | Aut(G) = #494587 (order 512) · Inn(G) = #99 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #99 |
| commutator, prime radical | C23 | C2×C8 |
| Frattini, cosocle | C2×M16 | C22 |
| socle | C2 | #324 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
45 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,9,5,10,15,23,6,8,14,20,16,21,27,30)(3,7,12,17,13,19,26,28,11,18,24,22,25,29,32,31);(1,3)(2,7,8,18)(4,12)(5,13)(6,11)(9,22,20,17)(10,19,21,29)(14,24)(15,26)(16,25)(23,31,30,28)(27,32)Polycyclic presentation:
2^7|a^2=d,b^2=e,d^2=f,f^2=g,b^a=bc,c^a=ce,e^a=eg,c^b=cg,d^b=de,d^c=dg26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1328, #11458, #11483, … |
| maximal normal subgroup of | C2×#1328, #11458, #11483, … |
| maximal quotient of | #7064, #7065, #7066, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 60);