Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) = #72237540 (order 1024) · Inn(G) = C42 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C42 |
| commutator | C4 | C2×C42 |
| Frattini, cosocle | C2×C8 | C23 |
| socle | C2 | C2×#97 |
| prime radical | C23 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,8,5,10,16,24,6,9,17,23,18,25,31,32)(3,11,13,27,14,20,29,22,12,26,15,30,28,7,19,21);(1,3)(2,7,9,20)(4,15)(5,14)(6,12)(8,22,23,21)(10,11,25,26)(13,17)(16,19)(18,28)(24,30,32,27)(29,31);(3,12)(7,20)(11,26)(13,15)(14,28)(19,29)(21,22)(27,30)Polycyclic presentation:
2^7|a^2=d,b^2=e,d^2=f,f^2=g,b^a=bf,e^a=eg,c^b=cg,d^b=dg44 conjugacy classes — view the full character table.
| maximal subgroup of | #11912, #11913, #11914, … |
| maximal normal subgroup of | #11912, #11913, #11914, … |
| maximal quotient of | #11405, #11406, #11413, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 842);