Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) = #7622868 (order 512) · Inn(G) = #99 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #99 |
| commutator | C2×C4 | C2×C8 |
| Frattini, cosocle | C2×M16 | C22 |
| socle | C2 | #323 |
| prime radical | C2×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,10,6,9,20,26,7,11,21,28,4,16,18,27)(3,12,14,29,13,8,22,23,15,25,19,31,17,24,30,32);(1,3)(2,8)(4,17)(5,19)(6,13)(7,15)(9,25)(10,23)(11,24)(12,16)(14,21)(18,22)(20,30)(26,31)(27,29)(28,32)Polycyclic presentation:
2^7|a^2=d,c^2=g,d^2=f,f^2=g,b^a=bc,c^a=ce,e^a=eg,c^b=cg,d^b=deg,d^c=dg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7346, #7347, #7377, … |
| maximal normal subgroup of | #7346, #7347, #7377, … |
| maximal quotient of | #7056, #7057, #7058, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 53);