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) = C23 |
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:
(2,5)(3,7)(9,10,13,16)(11,15,12,14)(17,18,22,20)(19,24,23,21);(1,2,4,5)(3,6,7,8)(9,11,13,12)(10,14,16,15)(17,19)(18,21)(20,24)(22,23);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,20,22,18)(19,24,23,21);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)(17,21,22,24)(18,23,20,19)Polycyclic presentation:
2^7|a^2=eg,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 | #26755, #26759, #26762, … |
| maximal normal subgroup of | #26755, #26759, #26762, … |
| maximal quotient of | #13885, #13933, #14006, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1241);