Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | Aut(G) = C24×D42 (order 1024) · Inn(G) = #121 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #121 |
| 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 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,17,20,24)(12,21,18,15)(14,22,23,19);(2,5)(6,7)(9,11,18,22,13,20,12,19)(10,14,21,24,16,23,15,17);(2,6)(4,8)(5,7)Polycyclic presentation:
2^7|a^2=f,b^2=dfg,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,e^c=eg29 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1683, #19110, #19126, … |
| maximal normal subgroup of | C2×#1683, #19110, #19126, … |
| maximal quotient of | #8648, #9215, #9219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 415);