Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | Aut(G) = #72237019 (order 4096) · Inn(G) = #381 · Out(G) = AΣL(1,𝔽16) |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #381 |
| commutator, prime radical | C23 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,6,9,19,24)(3,11,13,25,14,7,18,21)(4,15,17,29,16,8,22,23)(12,20,28,31,27,26,30,32);(1,3,6,14)(2,7,9,11)(4,12,16,27)(5,18,19,13)(8,20,15,26)(10,25,24,21)(17,30,22,28)(23,32,29,31);(1,4)(2,8,9,15)(3,12)(5,17)(6,16)(7,20,11,26)(10,23,24,29)(13,28)(14,27)(18,30)(19,22)(21,31,25,32)Polycyclic presentation:
2^7|a^2=f,b^2=g,c^2=d,f^2=g,b^a=bd,c^a=ce,d^a=dg,f^b=fg,e^c=eg29 conjugacy classes — view the full character table.
| maximal subgroup of | #12729, #12730, C2×#1830, … |
| maximal normal subgroup of | #12729, #12730, C2×#1830, … |
| maximal quotient of | #8735, #8737, #8753, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 562);