Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 6144 |
| 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 | |
668 conjugacy classes of subgroups; only the 384 normal subgroups are shown.
This group has 668 conjugacy classes of subgroups — too many to draw. Only its 384 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)(6,11)(10,13)(12,15)(14,16);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15);(2,7)(3,10)(8,15)(11,16);(2,7)(3,11)(6,13)(8,15)(9,14)(10,16);(2,8)(6,9)(7,15)(13,14)Polycyclic presentation:
2^7|b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df38 conjugacy classes — view the full character table.
| Fitting subgroup of | #3532⋊C3, #11714270, #11715753 |
| maximal subgroup of | #12753, #33551, C2×#3532, … |
| maximal normal subgroup of | #12753, #33551, C2×#3532, … |
| maximal quotient of | #34356, #34406, #34407, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2264);