Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = #60558 (order 256) · Inn(G) = C2×D4 · Out(G) = #41 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×D4 |
| commutator | C4 | C22×C8 |
| Frattini, cosocle | C2×C8 | C23 |
| socle | C2 | C8×D4 |
| prime radical | #135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,12,7,11,24,30,8,13,25,15,5,21,23,31)(3,14,17,19,16,9,26,20,10,27,29,18,22,28,32,4);(1,3)(2,9)(4,15)(5,22)(6,17)(7,16)(8,10)(11,27)(12,20)(13,28)(14,21)(18,30)(19,31)(23,32)(24,26)(25,29);(1,4,6,14,7,19,24,9,8,20,25,27,5,18,23,28)(2,10,12,29,11,22,30,32,13,3,15,17,21,16,31,26)Polycyclic presentation:
2^7|a^2=e,c^2=e,d^2=g,e^2=f,f^2=g,b^a=bd,c^a=cg,d^a=dg,d^b=dg44 conjugacy classes — view the full character table.
| maximal subgroup of | #12068, #12465, #12466, … |
| maximal normal subgroup of | #12068, #12465, #12466, … |
| maximal quotient of | #12038, #12049, #12050, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 903);