Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C22×C4×#120 (order 512) · Inn(G) = C23 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C23 |
| commutator, socle | C22 | C22×C8 |
| prime radical | C2×D4 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
81 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,9,13,10,7,14,16,8,11,17,19,18,15,12,20);(2,4)(6,11)(7,15)(13,19)(16,20);(5,7,9,16,10,11,14,19,8,15,17,20,18,6,12,13)Polycyclic presentation:
2^7|a^2=e,c^2=e,e^2=f,f^2=g,b^a=bd,c^a=cg,c^b=cg56 conjugacy classes — view the full character table.
| maximal subgroup of | #7380, #12038, #12039, … |
| maximal normal subgroup of | #7380, #12038, #12039, … |
| maximal quotient of | #12039, #12051, #12054, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 901);