Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) = C24×D42 (order 1024) · Inn(G) = #120 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C42 | C23 |
| prime radical | C2×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,14,18)(11,19,22,23)(12,15,13,17)(16,20,24,21);(1,2,5,7)(3,6,4,8)(10,15)(12,13)(16,23)(17,18)(19,24)(20,21);(1,3,5,4)(2,6,7,8)(9,11,14,22)(10,16,18,24)(12,21,13,20)(15,23,17,19)Polycyclic presentation:
2^7|a^2=g,b^2=fg,c^2=f,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,d^b=df,e^b=eg,d^c=dg,e^c=eg26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1652, #18503, #18521, … |
| maximal normal subgroup of | C2×#1652, #18503, #18521, … |
| maximal quotient of | #7775, #7776, #7783, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 384);