Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 12288 |
| 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 | |
561 conjugacy classes of subgroups; only the 390 normal subgroups are shown.
This group has 561 conjugacy classes of subgroups — too many to draw. Only its 390 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) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,4)(6,7)(10,13)(14,16);(1,2,5,4)(3,7,8,6);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,13)(15,16);(1,3,5,8)(2,6,4,7);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^7|b^2=f,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| Fitting subgroup of | #3517⋊C3, #11711038, #11712026, … |
| maximal subgroup of | #8414, #17655, #17684, … |
| maximal normal subgroup of | #8414, #17655, #17684, … |
| maximal quotient of | #34193, #34232, #34242, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2249);