Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | Aut(G) = #6346637 (order 512) · Inn(G) = #327 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #327 |
| commutator | C2×C8 | C2×C4 |
| Frattini, cosocle | C2×M16 | C22 |
| prime radical | C23 | D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,12,14,29)(4,17,19,28)(5,10)(6,22)(7,11)(8,13)(9,20,26,21)(15,25,16,32)(18,27)(23,30)(24,31);(1,3,7,14)(2,8,11,24)(4,18,19,30)(5,16,6,15)(9,25,26,32)(10,27,22,23)(12,28,29,17)(13,20,31,21)Polycyclic presentation:
2^7|a^2=d,b^2=g,c^2=fg,e^2=g,f^2=g,b^a=bc,c^a=ce,f^a=fg,c^b=cf,d^b=defg,e^b=eg,f^b=fg,d^c=dg20 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1340, #11685, #11769, … |
| maximal normal subgroup of | C2×#1340, #11685, #11769, … |
| maximal quotient of | #7078, #7079, #7082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 72);