Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | #532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
212 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 96 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 — 4 generators on 16 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,15)(14,16);(2,5)(6,8)(9,10,13,15,14,16,12,11);(1,3,4,7)(2,6,5,8)(9,11,12,16,14,15,13,10);(9,12,14,13)(10,15,16,11)Polycyclic presentation:
2^7|b^2=f,c^2=e,d^2=g,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,d^b=dg,d^c=dg32 conjugacy classes — view the full character table.
| maximal subgroup of | #9419, #9430, #9433, … |
| maximal normal subgroup of | #9419, #9430, #9433, … |
| maximal quotient of | #20764, #21093, #21200, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1873);