Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) = #495725 (order 512) · Inn(G) = C2×D8 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D8 |
| commutator | C8 | C22×C4 |
| Frattini, cosocle | C2×C8 | C23 |
| socle | C2 | C4×D8 |
| prime radical | #508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,11,7,13,23,12)(3,14,16,29,18,31,17,25)(4,10,19,28,20,9,5,21)(8,24,26,32,27,22,30,15);(1,3)(2,8)(4,15)(5,22)(6,16)(7,18)(9,25)(10,29)(11,26)(12,30)(13,27)(14,21)(17,23)(19,24)(20,32)(28,31);(1,4,6,19,7,20,23,5)(2,9,11,21,13,10,12,28)(3,15,16,24,18,32,17,22)(8,25,26,14,27,29,30,31)Polycyclic presentation:
2^7|a^2=e,c^2=e,d^2=fg,e^2=g,f^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fg,d^b=df,f^b=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7389, #12068, #13657, … |
| maximal normal subgroup of | #7389, #12068, #13657, … |
| maximal quotient of | #12043, #12044, #12045, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 911);