Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = #72232191 (order 2048) · Inn(G) = C2×D4 · Out(G) = C2×D42 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C22 | C2×C22⋊C4 |
| prime radical | C24 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,15,13,11,16,14);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,14)(15,16)Polycyclic presentation:
2^7|a^2=f,c^2=d,e^2=g,f^2=g,b^a=bd,c^a=ce,e^a=eg,e^c=eg38 conjugacy classes — view the full character table.
| maximal subgroup of | #8360, #8362, #8363, … |
| maximal normal subgroup of | #8360, #8362, #8363, … |
| maximal quotient of | #8706, #8707, #8708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 558);