Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | Aut(G) = #72236189 (order 1024) · Inn(G) = #392 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #392 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
190 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 44 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 — 3 generators on 16 points:
(1,2)(3,11)(4,9)(5,8)(6,10)(7,12)(13,16)(14,15);(1,3)(2,7)(4,14)(5,12)(6,13)(8,16)(9,11)(10,15);(2,8)(4,5)(7,11)(9,10)(12,14)(15,16)Polycyclic presentation:
2^7|d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cf,d^b=dg,d^c=dg,e^c=eg23 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2008, #23412, #23872, … |
| maximal normal subgroup of | C2×#2008, #23412, #23872, … |
| maximal quotient of | #9930, #9932, #9934, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 740);