Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | Aut(G) = C2×#6342371 (order 1024) · Inn(G) = C24 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C24 |
| commutator, socle | C22 | C23×C4 |
| prime radical | #121 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
173 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 173 conjugacy classes of subgroups — too many to draw. Only its 124 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(9,10,12,16,13,15,19,22)(11,14,18,21,17,20,23,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24);(2,5)(6,8)(10,15)(11,17)(16,22)(18,23);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)Polycyclic presentation:
2^7|a^2=f,d^2=e,f^2=g,b^a=be,c^a=cg,d^a=de,c^b=ceg,d^b=dg38 conjugacy classes — view the full character table.
| maximal subgroup of | #8916, #8917, #8997, … |
| maximal normal subgroup of | #8916, #8917, #8997, … |
| maximal quotient of | #20551, #20554, #20563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1727);