Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,19)(12,15)(13,23)(14,18)(16,21)(17,20)(22,24);(2,5)(6,8)(9,11,20,22,14,21,13,12)(10,15,23,16,18,24,17,19);(1,3,4,7)(2,6,5,8)(9,12,13,21,14,22,20,11)(10,16,17,15,18,19,23,24);(11,21)(12,22)(15,24)(16,19)Polycyclic presentation:
2^7|b^2=fg,c^2=e,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,d^b=dg,d^c=dg32 conjugacy classes — view the full character table.
| maximal subgroup of | #9439, #10022, #10024, … |
| maximal normal subgroup of | #9439, #10022, #10024, … |
| maximal quotient of | #20765, #21094, #21095, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1872);