Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) = #72213344 (order 2048) · Inn(G) = C42 · Out(G) = #2021 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C42 |
| commutator | C4 | C4×C8 |
| Frattini, cosocle | C2×C42 | C22 |
| socle | C22 | #95 |
| prime radical | C23 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,5,4,6,7,8)(9,10)(11,17,19,23)(12,15)(13,16)(14,18,21,24)(20,22);(9,11)(10,14,16,21)(12,18)(13,19)(15,17,22,23)(20,24)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=f,d^2=g,b^a=bc,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| maximal subgroup of | #7115, #7129, C2×#1274, … |
| maximal normal subgroup of | #7115, #7129, C2×#1274, … |
| maximal quotient of | #7014, #7017, #7018, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 6);