Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | D4 |
| commutator | C4 | C2×C42 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C23 | C2×D4 |
| prime radical | C24 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23);(2,4)(9,11,13,21,14,22,12,20)(10,15,16,23,18,24,17,19);(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,24,19)Polycyclic presentation:
2^7|a^2=e,b^2=f,c^2=g,d^2=g,f^2=g,b^a=bd,d^a=dg,f^a=fg56 conjugacy classes — view the full character table.
| maximal subgroup of | #7260, #7262, #10706, … |
| maximal normal subgroup of | #7260, #7262, #10706, … |
| maximal quotient of | #7872, #7875, #7878, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 467);