Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) = C24×#120 (order 512) · Inn(G) = #120 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | #121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
104 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,8,18)(3,12,23,32)(4,6,11,7)(5,17,24,28)(9,25,31,14)(10,21,20,22)(13,29,26,19)(15,27,16,30);(1,3)(2,9)(4,13)(5,19)(6,16)(7,15)(8,23)(10,27)(11,26)(12,17)(14,22)(18,31)(20,30)(21,25)(24,29)(28,32);(1,4,7,2)(3,13,15,9)(5,20,22,28)(6,18,8,11)(10,21,17,24)(12,29,27,25)(14,32,19,30)(16,31,23,26)Polycyclic presentation:
2^7|a^2=fg,c^2=f,d^2=f,b^a=bd,c^a=ce,d^a=df,d^b=df,e^b=eg,d^c=dg23 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1615, #18107, #18127, … |
| maximal normal subgroup of | C2×#1615, #18107, #18127, … |
| maximal quotient of | #7787, #7788, #7798, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 347);