Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | Aut(G) = #72236189 (order 1024) · Inn(G) = C2×#121 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#121 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
233 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 233 conjugacy classes of subgroups — too many to draw. Only its 98 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 — 4 generators on 16 points:
(2,6)(3,9)(4,12)(8,14)(10,11)(13,16);(1,2,4,6,5,8,12,14)(3,7,10,13,11,15,9,16);(1,3)(2,7)(4,10)(5,11)(6,13)(8,15)(9,12)(14,16);(2,8)(6,14)(7,15)(13,16)Polycyclic presentation:
2^7|b^2=e,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,d^b=dg,f^c=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | #24420, #24422, #24472, … |
| maximal normal subgroup of | #24420, #24422, #24472, … |
| maximal quotient of | #19162, #19165, #19166, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1800);