Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C23 |
| commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C24 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18,20,22)(19,21,23,24);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,21)(20,23)(22,24);(10,14)(13,16)(19,23)(21,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^7|a^2=fg,b^2=f,d^2=e,b^a=be,c^a=cf,c^b=cg44 conjugacy classes — view the full character table.
| Fitting subgroup of | #2401⋊C3 |
| maximal subgroup of | #21668, #21669, #21670, … |
| maximal normal subgroup of | #21668, #21669, #21670, … |
| maximal quotient of | #13842, #13846, #13855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1133);