Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | Aut(G) = #72236027 (order 2048) · Inn(G) = #121 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #121 |
| commutator, Frattini, cosocle | C42 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8)(3,9,11,16)(4,13,10,7)(6,14,15,12)(17,18,21,24)(19,25,27,32)(20,28,30,23)(22,29,31,26);(1,3,10,14,5,11,4,12)(2,6,13,16,8,15,7,9);(3,11)(6,15)(9,16)(12,14)(17,19,21,27)(18,22,24,31)(20,29,30,26)(23,32,28,25)Polycyclic presentation:
2^7|a^2=fg,b^2=df,c^2=g,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,c^b=cf,e^c=eg23 conjugacy classes — view the full character table.
| maximal subgroup of | #12388, #12389, #12412, … |
| maximal normal subgroup of | #12388, #12389, #12412, … |
| maximal quotient of | #8653, #9224, #9576, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 428);