Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| socle | C22 | C22×D4 |
| prime radical | #532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
196 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)(11,20)(12,16)(13,18)(14,23)(15,19)(17,22)(21,24);(2,5)(6,8)(9,11,14,12,15,22,13,21)(10,16,18,20,19,24,23,17);(1,3,4,7)(2,6,5,8)(9,12,14,22,15,21,13,11)(10,17,18,16,19,20,23,24);(9,13,15,14)(10,18,19,23)(11,21,22,12)(16,20,24,17)Polycyclic presentation:
2^7|b^2=f,c^2=e,d^2=g,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,c^b=cg32 conjugacy classes — view the full character table.
| Frattini subgroup of | #94784, #94786 |
| maximal subgroup of | #9005, #9008, #10033, … |
| maximal normal subgroup of | #9005, #9008, #10033, … |
| maximal quotient of | #20762, #21086, #21199, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1867);