Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C25 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
336 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 104 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:
(2,4)(5,6,8,10)(7,11,12,9)(14,17)(15,19);(5,7,8,12)(6,9,10,11)(13,14,16,17)(15,18,19,20);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11);(5,6)(7,9)(8,10)(11,12)(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^7|a^2=g,b^2=e,b^a=be,c^a=cf,d^a=de,c^b=cg38 conjugacy classes — view the full character table.
| maximal subgroup of | #9477, #9481, #9512, … |
| maximal normal subgroup of | #9477, #9481, #9512, … |
| maximal quotient of | #13868, #13958, #13991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1214);