Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | Aut(G) order 4096 |
| 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 ◁ #117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
185 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(9,10)(11,14)(12,15)(13,17)(16,20)(18,22)(19,21)(23,24);(1,2)(3,6)(4,5)(7,8)(9,11,13,18)(10,14,17,22)(12,19,20,24)(15,21,16,23);(2,5)(6,8)(10,15)(12,20)(14,21)(16,17)(19,24)(22,23);(1,3,4,7)(2,6,5,8)(11,18)(14,22)(19,24)(21,23)Polycyclic presentation:
2^7|b^2=g,d^2=e,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 | #12164, #12167, #12170, … |
| maximal normal subgroup of | #12164, #12167, #12170, … |
| maximal quotient of | #21897, #21968, #22126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1991);