Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator | C4 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| prime radical | C2×#135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,11,16)(7,17,8,13)(9,15,10,14)(12,19,20,18);(5,7,11,8)(6,12,16,20)(9,18,10,19)(13,14,17,15);(1,3,4,2)(5,8,11,7)(6,13,16,17)(9,18,10,19)(12,14,20,15);(5,9,11,10)(6,14,16,15)(7,18,8,19)(12,17,20,13)Polycyclic presentation:
2^7|a^2=f,b^2=g,c^2=f,d^2=g,e^2=g,b^a=be,d^a=dg,e^a=eg,e^b=eg,d^c=dg44 conjugacy classes — view the full character table.
| maximal subgroup of | #12828, #12949, #13030, … |
| maximal normal subgroup of | #12828, #12949, #13030, … |
| maximal quotient of | #17858, #17882, #17899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1684);