Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) = #72237167 (order 2048) · Inn(G) = C2×D8 · Out(G) = #525 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| prime radical | C2×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
71 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,20)(12,17)(13,16)(14,22)(15,19)(18,21)(23,24);(2,5)(6,8)(9,10,13,17,14,18,20,24,15,19,23,11,21,22,12,16);(1,3,4,7)(2,6,5,8)(9,11,12,19,21,24,23,18,15,17,20,10,14,16,13,22)Polycyclic presentation:
2^7|b^2=e,c^2=d,d^2=fg,e^2=f,f^2=g,b^a=bd,c^a=ce,d^a=df,e^a=efg,f^a=fg,c^b=cg26 conjugacy classes — view the full character table.
| maximal subgroup of | #12376, #12377, #12378, … |
| maximal normal subgroup of | #12376, #12377, #12378, … |
| maximal quotient of | #12890, #13149, #13150, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 975);