Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C2×D4 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
171 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 171 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(1,2,5,4)(3,7,8,6)(9,10,12,16)(11,18,19,14)(13,15,21,17)(20,22,24,23);(1,3)(2,6)(4,7)(5,8)(9,11,12,19)(10,14,16,18)(13,20,21,24)(15,23,17,22);(1,4,5,2)(3,7,8,6)(10,15)(11,19)(13,21)(14,22)(16,17)(18,23);(3,8)(6,7)(10,16)(11,19)(15,17)(20,24)Polycyclic presentation:
2^7|a^2=e,b^2=g,c^2=eg,f^2=g,b^a=be,c^a=cf,d^a=dg,f^a=fg,c^b=cg,d^b=de,f^c=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | #19865, #19866, #19881, … |
| maximal normal subgroup of | #19865, #19866, #19881, … |
| maximal quotient of | #22502, #22504, #22623, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2088);