Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) = #72238380 (order 8192) · Inn(G) = C2×C22⋊C4 · Out(G) = #23900 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C24 | C23 |
| prime radical | C25 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
252 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)(3,9,11,7)(4,6)(5,8)(10,15,16,13)(12,14);(2,6)(7,13)(8,14)(9,15);(1,3,4,10)(2,7,6,13)(5,11,12,16)(8,9,14,15)Polycyclic presentation:
2^7|a^2=f,c^2=d,b^a=bd,c^a=ce,e^a=eg,f^c=fg32 conjugacy classes — view the full character table.
| Frattini subgroup of | #94530 |
| maximal subgroup of | #7213, #7349, #8535, … |
| maximal normal subgroup of | #7213, #7349, #8535, … |
| maximal quotient of | #8698, #8700, #8702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 551);