Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | Aut(G) = #60558 (order 256) · Inn(G) = #102 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #102 |
| commutator | C8 | C2×C8 |
| Frattini, cosocle | C4×C8 | C22 |
| socle | C2 | D4⋊C8 |
| prime radical | #135 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,5,9,6,10,16,15,7,11,4,13,14,3,12,8);(1,3)(2,6)(4,8)(5,15)(7,10)(9,12)(11,14)(13,16)Polycyclic presentation:
2^7|a^2=d,c^2=e,d^2=f,e^2=g,f^2=g,b^a=bc,c^a=ce,e^a=eg,c^b=ceg,d^b=dg,e^b=eg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1336, #11604, #11669, … |
| maximal normal subgroup of | C2×#1336, #11604, #11669, … |
| maximal quotient of | #7070, #7071, #7072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 68);