Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
186 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2)(3,6)(4,5)(7,8)(9,10,14,16)(11,17,18,24)(12,13,20,21)(15,22,23,19);(2,5)(6,8)(9,11,13,19,14,18,21,22)(10,15,12,17,16,23,20,24);(1,3,4,7)(2,6,5,8)(9,12,13,16,14,20,21,10)(11,17,19,23,18,24,22,15);(11,18)(15,23)(17,24)(19,22)Polycyclic presentation:
2^7|a^2=g,b^2=f,c^2=e,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,d^b=dg32 conjugacy classes — view the full character table.
| maximal subgroup of | #26849, #26851, #26852, … |
| maximal normal subgroup of | #26849, #26851, #26852, … |
| maximal quotient of | #20763, #21087, #21088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1869);