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) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
211 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 211 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,15)(13,17)(16,20)(18,22)(19,21)(23,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,19)(13,18)(15,21)(16,23)(17,22)(20,24);(3,8)(5,7)(10,15)(12,20)(14,21)(16,17)(19,24)(22,23);(2,6)(5,7)(11,18)(14,22)(19,24)(21,23)Polycyclic presentation:
2^7|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×#3198, #60749, #60753, … |
| maximal normal subgroup of | C2×#3198, #60749, #60753, … |
| maximal quotient of | #21770, #21772, #21945, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1930);