Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 12288 |
| 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×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(6,10)(8,11)(12,16)(13,17)(15,18)(19,20);(5,6,11,17,9,13,8,10)(7,12,18,20,14,19,15,16);(5,7)(6,12)(8,15)(9,14)(10,16)(11,18)(13,19)(17,20);(7,14)(12,19)(15,18)(16,20)Polycyclic presentation:
2^7|a^2=f,b^2=eg,e^2=g,b^a=be,e^a=eg,d^c=dg44 conjugacy classes — view the full character table.
| Fitting subgroup of | #2913⋊C3, #11711549, #11711550, … |
| maximal subgroup of | #12925, #12926, #12996, … |
| maximal normal subgroup of | #12925, #12926, #12996, … |
| maximal quotient of | #17789, #17799, #17812, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1645);