Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | Aut(G) = #72216601 (order 1024) · Inn(G) = #353 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #353 |
| commutator | C42 | C2×C4 |
| Frattini, cosocle | C4⋊Q8 | C22 |
| prime radical | #127 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,8,5,9,12,16)(4,6,10,14,13,7,11,15);(2,6)(3,10)(4,13)(7,9)(11,12)(14,15)Polycyclic presentation:
2^7|a^2=d,c^2=fg,d^2=g,e^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,c^b=cfg,d^b=def,e^b=eg,d^c=dfg,e^d=eg17 conjugacy classes — view the full character table.
| maximal subgroup of | #7517, #7518, #7527, … |
| maximal normal subgroup of | #7517, #7518, #7527, … |
| maximal quotient of | #7354, #7371, #7404, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 142);