Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | Aut(G) order 1024 |
| 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 | #449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
188 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,7)(4,6)(5,8)(10,15)(11,18)(12,14)(16,23)(19,24)(20,22);(1,3,4,8)(2,5,6,7)(9,10)(11,19)(12,21)(13,15)(14,17)(16,20)(18,23)(22,24);(3,8)(5,7)(9,11)(10,16)(12,18)(13,20)(14,22)(15,19)(17,23)(21,24);(2,6)(5,7)(9,12,13,14)(10,17,15,21)(11,18,20,22)(16,23,19,24)Polycyclic presentation:
2^7|b^2=eg,d^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,c^b=ce,d^b=dg,f^c=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#3225, #60797, #60807, … |
| maximal normal subgroup of | C2×#3225, #60797, #60807, … |
| maximal quotient of | #21814, #21871, #21992, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1957);