Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) = #72233347 (order 4096) · Inn(G) = C2×C22⋊C4 · Out(G) = #1790 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C42 | C23 |
| prime radical | C23 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18)(19,25,27,32)(20,24)(21,23)(22,26,30,28)(29,31);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,22)(20,28)(21,27)(23,30)(24,32)(25,31)(26,29);(18,23)(22,30)(24,31)(25,32)Polycyclic presentation:
2^7|a^2=e,b^2=f,d^2=g,e^2=f,b^a=bd,c^a=cg,d^a=df,d^b=dg,e^b=efg26 conjugacy classes — view the full character table.
| maximal subgroup of | #8525, #8526, #8527, … |
| maximal normal subgroup of | #8525, #8526, #8527, … |
| maximal quotient of | #7593, #7602, #7641, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 266);