Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) = C2×C4×#120 (order 256) · Inn(G) = C22⋊C4 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22⋊C4 |
| commutator | C2×C4 | C2×C8 |
| Frattini, cosocle | C4×C8 | C22 |
| socle, prime radical | C22 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
43 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,12,17,14,15,20,22,13,16,21,23,11,18,19,24);(1,2,4,6)(3,8,7,5)(9,11,13,14)(10,15,16,18)(12,20,21,19)(17,24,23,22)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=eg,d^2=f,f^2=g,b^a=bc,c^a=ce,c^b=ceg,d^b=dg38 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1338, #11610, #11611, … |
| maximal normal subgroup of | C2×#1338, #11610, #11611, … |
| maximal quotient of | #7118, #7128, #7170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 70);