Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C23 |
| commutator, socle | C23 | C22×C4 |
| prime radical | C24 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(6,9)(10,12)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(2,4)(5,7,8,11)(6,10,9,12)(15,21)(18,24)(22,27)(25,28)Polycyclic presentation:
2^7|a^2=f,c^2=d,f^2=g,b^a=bd,c^a=ce,c^b=cg44 conjugacy classes — view the full character table.
| Frattini subgroup of | #93203, #93209, #93210, … |
| maximal subgroup of | #7059, #7325, #7327, … |
| maximal normal subgroup of | #7059, #7325, #7327, … |
| maximal quotient of | #8702, #8703, #8704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 550);