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) = C22 |
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 | |
79 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,10,7,12,6,11)(3,13,15,28,17,30,16,29)(4,18,19,27,20,9,21,25)(8,22,23,14,24,31,26,32);(1,3,6,16,7,17,5,15)(2,8,11,23,12,24,10,26)(4,14,19,31,20,32,21,22)(9,13,25,29,18,30,27,28)Polycyclic presentation:
2^7|a^2=d,b^2=e,d^2=g,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 | #7400, #7401, #7407, … |
| maximal normal subgroup of | #7400, #7401, #7407, … |
| maximal quotient of | #7188, #7203, #7216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 124);