Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | Aut(G) = #72236189 (order 1024) · Inn(G) = #386 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #386 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
155 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 155 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10,7,12,5,11)(3,13,16,23,17,26,15,8)(4,18,20,9,21,24,19,25)(14,22,29,31,30,32,28,27);(1,3)(2,8,12,23)(4,14)(5,15)(6,16)(7,17)(9,22,25,32)(10,13,11,26)(18,27,24,31)(19,28)(20,29)(21,30);(1,4,7,21)(2,9,12,25)(3,14,17,30)(5,20,6,19)(8,22,23,32)(10,18,11,24)(13,27,26,31)(15,29,16,28)Polycyclic presentation:
2^7|a^2=f,b^2=df,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^b=eg,d^c=dg,e^c=eg,f^c=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1905, #20998, #22284, … |
| maximal normal subgroup of | C2×#1905, #20998, #22284, … |
| maximal quotient of | #9088, #9090, #9092, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 637);