Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C25 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
496 conjugacy classes of subgroups; only the 392 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 392 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(5,8)(9,10,13,16)(11,14,19,17)(12,20,21,15)(18,23,24,22);(1,2,4,6)(3,5,7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24);(1,3,4,7)(2,5,6,8);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)Polycyclic presentation:
2^7|a^2=f,b^2=g,c^2=g,e^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| Frattini subgroup of | #94452 |
| maximal subgroup of | #8427, #8474, #12724, … |
| maximal normal subgroup of | #8427, #8474, #12724, … |
| maximal quotient of | #34188, #34209, #34214, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2241);