Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#393988 (order 1024) · Inn(G) = C22 · Out(G) = C2×#2010 |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C22 |
| commutator | C2 | C4×C16 |
| socle, prime radical | C23 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,4)(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24);(2,4)(5,6,7,8)Polycyclic presentation:
2^7|a^2=d,b^2=e,d^2=f,f^2=g,b^a=bc80 conjugacy classes — view the full character table.
| maximal subgroup of | #7102, #7112, #7113, … |
| maximal normal subgroup of | #7102, #7112, #7113, … |
| maximal quotient of | #7136, #7142, #7145, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 106);