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 | #538 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
120 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 120 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) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,14,17)(11,18,20,23)(12,16,13,21)(15,22,24,19);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(3,7)(5,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)Polycyclic presentation:
2^7|a^2=fg,c^2=fg,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×#1618, #18121, #18127, … |
| maximal normal subgroup of | C2×#1618, #18121, #18127, … |
| maximal quotient of | #7790, #7793, #7796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 350);