Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C24 |
| commutator | C2 | C22×C42 |
| socle | C22 | C23×C4 |
| prime radical | C24 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,12,9,11,15,18)(7,10,14,17,13,16,19,20);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20);(7,13)(10,16)(14,19)(17,20);(6,11)(10,16)(12,18)(17,20)Polycyclic presentation:
2^7|a^2=e,b^2=f,e^2=g,d^a=dg,c^b=cg68 conjugacy classes — view the full character table.
| Frattini subgroup of | #72292729 |
| maximal subgroup of | #9387, #9388, #9491, … |
| maximal normal subgroup of | #9387, #9388, #9491, … |
| maximal quotient of | #17314, #17319, #17324, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1605);