Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | Aut(G) = C2×#7622880 (order 1024) · Inn(G) = #120 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #120 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #386 |
| prime radical | #525 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
171 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,5,9,6,3,7,11)(4,10,13,16,12,8,14,15);(1,3,5,11,6,2,7,9)(4,10,14,15,12,8,13,16);(1,4)(2,8)(3,10)(5,14)(6,12)(7,13)(9,16)(11,15)Polycyclic presentation:
2^7|a^2=f,b^2=df,f^2=g,b^a=bd,c^a=ce,e^a=eg,e^b=eg,d^c=dg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #12755, #12756, #12757, … |
| maximal normal subgroup of | #12755, #12756, #12757, … |
| maximal quotient of | #8965, #8966, #8967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 623);