Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| 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 | #535 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
205 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 205 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)(9,10,13,17)(11,14,18,22)(12,15,20,16)(19,21,24,23);(1,3)(2,5)(4,8)(6,7)(9,11,13,18)(10,14,17,22)(12,19,20,24)(15,21,16,23);(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|a^2=g,b^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 | #13351, #13354, #13355, … |
| maximal normal subgroup of | #13351, #13354, #13355, … |
| maximal quotient of | #21770, #21771, #21946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1932);