Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = #72260440 (order 6144) · Inn(G) = C23 · Out(G) = C2×#73084 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C42 | C23 |
| commutator, socle | C22 | C2×C42 |
| prime radical | C23 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3,6,4,5,7,8);(9,10,11,14,12,13,15,16);(2,5)(6,8)(10,13)(14,16)Polycyclic presentation:
2^7|a^2=d,b^2=e,d^2=f,e^2=g,c^a=cf,c^b=cg56 conjugacy classes — view the full character table.
| Fitting subgroup of | #1455⋊C3, #11710956, #11711355 |
| maximal subgroup of | #7225, #8803, #8804, … |
| maximal normal subgroup of | #7225, #8803, #8804, … |
| maximal quotient of | #7846, #7852, #8136, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 187);