Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) = C24×D42 (order 1024) · Inn(G) = #120 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C42 | C23 |
| prime radical | C2×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
97 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,9)(3,10,12,15)(4,8,5,13)(7,14,16,11)(17,18,22,26)(19,27,30,31)(20,25,21,24)(23,29,32,28);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,22,30)(18,23,26,32)(20,29,21,28)(24,31,25,27);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,24)(20,21)(23,31)(25,26)(27,32)(28,29)Polycyclic presentation:
2^7|a^2=f,b^2=g,c^2=fg,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,d^b=df,e^b=eg,d^c=dg,e^c=eg26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1638, #18494, #18521, … |
| maximal normal subgroup of | C2×#1638, #18494, #18521, … |
| maximal quotient of | #7765, #7767, #7773, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 370);