Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) = #72237167 (order 2048) · Inn(G) = C2×D8 · Out(G) = #525 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6)(3,7,8,5)(10,14)(11,16)(12,18)(15,20)(17,24)(19,23)(21,22);(1,3,4,8)(2,5,6,7)(10,15)(14,20)(17,21)(22,24);(9,10,16,22,12,17,19,20,13,15,23,24,18,21,11,14)Polycyclic presentation:
2^7|a^2=d,b^2=d,c^2=ef,e^2=fg,f^2=g,b^a=bd,c^a=ce,e^a=ef,f^a=fg,c^b=cg26 conjugacy classes — view the full character table.
| maximal subgroup of | #12400, #12401, #12402, … |
| maximal normal subgroup of | #12400, #12401, #12402, … |
| maximal quotient of | #12908, #13167, #13168, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 987);