Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
366 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(6,10)(8,15)(11,17)(12,16)(14,19)(18,20);(5,6,9,12)(7,11,13,18)(8,16,15,10)(14,20,19,17);(2,4)(5,7)(6,11)(8,14)(9,13)(10,17)(12,18)(15,19)(16,20);(7,13)(11,18)(14,19)(17,20)Polycyclic presentation:
2^7|b^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,e^b=eg,d^c=dg32 conjugacy classes — view the full character table.
| maximal subgroup of | #13098, #13103, #23894, … |
| maximal normal subgroup of | #13098, #13103, #23894, … |
| maximal quotient of | #18226, #18234, #18242, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1742);