Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C23 |
| commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,11)(7,12,8,10)(13,14,16,18)(15,19,20,17);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,16,20)(14,17,18,19);(2,4)(14,18)(17,19);(5,8,9,7)(6,10,11,12)Polycyclic presentation:
2^7|a^2=e,b^2=e,d^2=ef,b^a=be,c^a=cf,c^b=cg44 conjugacy classes — view the full character table.
| maximal subgroup of | #9029, #9032, #10013, … |
| maximal normal subgroup of | #9029, #9032, #10013, … |
| maximal quotient of | #13842, #13847, #13850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1130);