Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| socle | C22 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
250 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 100 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)(5,6)(7,14)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19);(5,7,9,17,10,15,8,16)(6,11,12,14,13,19,18,20);(2,4)(7,15)(11,19)(14,20)(16,17);(5,8,10,9)(6,12,13,18)(7,16,15,17)(11,14,19,20)Polycyclic presentation:
2^7|b^2=e,d^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,c^b=cg32 conjugacy classes — view the full character table.
| maximal subgroup of | #11685, #11688, #12998, … |
| maximal normal subgroup of | #11685, #11688, #12998, … |
| maximal quotient of | #19133, #19137, #19142, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1786);