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 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
664 conjugacy classes of subgroups; only the 392 normal subgroups are shown.
This group has 664 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) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16);(1,3,5,11)(2,6,7,13)(4,10,12,16)(8,9,14,15);(2,7)(6,13)(8,14)(9,15);(3,10)(6,9)(11,16)(13,15);(2,8)(6,9)(7,14)(13,15)Polycyclic presentation:
2^7|b^2=g,b^a=bf,c^a=cg,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| Frattini subgroup of | #94453 |
| maximal subgroup of | #8346, #8535, #12725, … |
| maximal normal subgroup of | #8346, #8535, #12725, … |
| maximal quotient of | #34176, #34198, #34218, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2222);