Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | Aut(G) = C2×#511819 (order 1024) · Inn(G) = #127 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #127 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #392 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
197 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,9)(4,12)(5,8)(6,10)(7,14)(11,16)(13,15);(1,3)(2,6)(4,13)(5,11)(7,16)(8,15)(9,12)(10,14);(6,15)(9,16)Polycyclic presentation:
2^7|d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fg,c^b=cf,d^b=dg,e^b=eg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7396, #7397, #12114, … |
| maximal normal subgroup of | #7396, #7397, #12114, … |
| maximal quotient of | #9938, #9939, #9940, … |
| automorphism group of | #347 |
| outer automorphism group of | #2244, #2245, #2250, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 742);