Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | Aut(G) = #72231746 (order 1024) · Inn(G) = #99 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #99 |
| commutator | C23 | C42 |
| Frattini, prime radical, cosocle | C2×#41 | C22 |
| socle | C2 | #342 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
83 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,11)(4,15,16,8)(5,9)(6,10)(7,12,19,25)(13,23)(14,24)(17,28,29,21)(18,22)(20,26,30,32)(27,31);(1,3,5,13,6,14,18,27)(2,7,9,20,10,19,22,30)(4,12,17,26,16,25,29,32)(8,11,21,23,15,24,28,31)Polycyclic presentation:
2^7|a^2=d,b^2=e,e^2=g,b^a=bc,c^a=cf,f^a=fg,d^b=df,d^c=dg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7398, #7399, #7406, … |
| maximal normal subgroup of | #7398, #7399, #7406, … |
| maximal quotient of | #7188, #7204, #7214, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 123);