Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C2×#127 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,3)(6,7)(11,14)(12,16);(1,2,4,3)(5,6,8,7)(9,10,13,15)(11,12,14,16);(5,6,8,7)(9,11,13,14)(10,12,15,16);(1,3,4,2)(5,7,8,6)(9,12)(10,14)(11,15)(13,16)Polycyclic presentation:
2^7|b^2=g,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=dg44 conjugacy classes — view the full character table.
| maximal subgroup of | #7659, #7679, #7699, … |
| maximal normal subgroup of | #7659, #7679, #7699, … |
| maximal quotient of | #14687, #14700, #14708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1590);