Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#23794 (order 512) · Inn(G) = C22 · Out(G) = C2×#525 |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C16 | C22 |
| commutator | C2 | C4×C16 |
| socle, prime radical | C22 | C2×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 36 points:
(5,6,7,12,8,13,15,24,9,14,16,25,18,26,27,34,10,11,17,21,19,22,28,31,20,23,29,32,30,33,35,36);(1,2,3,4)(6,11)(12,21)(13,22)(14,23)(24,31)(25,32)(26,33)(34,36)Polycyclic presentation:
2^7|a^2=c,b^2=d,c^2=e,e^2=f,f^2=g,b^a=bg80 conjugacy classes — view the full character table.
| Frattini subgroup of | #94506, #94509, #94613, … |
| maximal subgroup of | #7330, #7332, #7380, … |
| maximal normal subgroup of | #7330, #7332, #7380, … |
| maximal quotient of | #7329, #7331, #7465, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 129);