Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 8192 |
| 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 | #534 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
526 conjugacy classes of subgroups; only the 388 normal subgroups are shown.
This group has 526 conjugacy classes of subgroups — too many to draw. Only its 388 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) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(5,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24);(1,2,4,6)(3,5,7,8)(10,14)(15,21)(16,19)(22,23);(1,3)(2,5)(4,7)(6,8)(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23);(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24);(9,12)(10,16)(11,17)(13,20)(14,19)(15,22)(18,24)(21,23)Polycyclic presentation:
2^7|b^2=g,c^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| Frattini subgroup of | #94454 |
| maximal subgroup of | #8348, #8540, #12726, … |
| maximal normal subgroup of | #8348, #8540, #12726, … |
| maximal quotient of | #34177, #34199, #34219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2223);