Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | Aut(G) = C2×#72233929 (order 4096) · Inn(G) = #127 · Out(G) = C2×#392 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #127 |
| commutator, Frattini, cosocle | C42 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,13,16)(11,17,20,24)(12,21,18,15)(14,22,23,19);(1,2,3,5,4,6,7,8);(2,6)(5,8)(9,11,18,22,13,20,12,19)(10,14,21,24,16,23,15,17)Polycyclic presentation:
2^7|a^2=g,b^2=d,c^2=eg,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,c^b=cf26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1723, #19618, #19619, … |
| maximal normal subgroup of | C2×#1723, #19618, #19619, … |
| maximal quotient of | #8820, #9833, #9834, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 455);