Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
190 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,3)(4,7,8,6)(10,12)(11,15)(18,22)(20,23);(2,3)(6,7)(9,10,13,12)(11,14,15,16)(17,18)(19,23)(20,24)(21,22);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,16)(13,15)(17,19,21,24)(18,23,22,20);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,16,15,14)(17,20,21,23)(18,24,22,19)Polycyclic presentation:
2^7|a^2=g,b^2=f,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=cg38 conjugacy classes — view the full character table.
| maximal subgroup of | #26753, #26757, #26760, … |
| maximal normal subgroup of | #26753, #26757, #26760, … |
| maximal quotient of | #13885, #13918, #13985, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1240);