Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#2929 (order 256) · Inn(G) = C22 · Out(G) = C2×#131 |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C22⋊C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C16 | C22 |
| commutator | C2 | C2×C32 |
| socle | C22 | C2×C16 |
| prime radical | C23 | C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
41 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2)(3,4)(5,6,7,11,8,12,15,21,9,13,16,22,18,24,27,31,10,14,17,23,19,25,28,32,20,26,29,33,30,34,35,36);(2,4)Polycyclic presentation:
2^7|a^2=d,d^2=e,e^2=f,f^2=g,b^a=bc80 conjugacy classes — view the full character table.
| maximal subgroup of | #7334, #7335, #7362, … |
| maximal normal subgroup of | #7334, #7335, #7362, … |
| maximal quotient of | #7334, #7362, D4⋊C32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 131);