Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | Aut(G) = #72231669 (order 1024) · Inn(G) = #102 · Out(G) = C25 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #102 |
| commutator | C8 | C42 |
| Frattini, cosocle | C22×C8 | C22 |
| prime radical | C23 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4)(5,6,9,13)(7,8,15,17)(10,12,18,20)(11,19,16,14);(5,7,10,14,11,12,17,13,9,15,18,19,16,20,8,6)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=fg,e^2=f,f^2=g,b^a=bc,c^a=cf,e^a=ef,f^a=fg,c^b=cg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1386, #12567, C4×#360, … |
| maximal normal subgroup of | C2×#1386, #12567, C4×#360, … |
| maximal quotient of | #7158, #7175, #7178, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 118);