Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) = #72231575 (order 2048) · Inn(G) = D8 · Out(G) = #2388 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D8 |
| commutator | C8 | C42 |
| Frattini, cosocle | C22×C8 | C22 |
| socle | C22 | #107 |
| prime radical | C23 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2)(3,4)(5,6,9,14,11,17,23,16)(7,18,20,32,22,33,34,27)(8,15,10,13,24,30,25,31)(12,21,26,35,29,36,28,19);(2,4)(5,7,10,21,9,20,24,35,11,22,25,36,23,34,8,19)(6,12,15,27,16,28,31,33,17,29,30,32,14,26,13,18)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=fg,d^2=g,e^2=f,f^2=g,b^a=bc,c^a=cf,e^a=efg,f^a=fg44 conjugacy classes — view the full character table.
| maximal subgroup of | #7420, #7421, C2×#1381, … |
| maximal normal subgroup of | #7420, #7421, C2×#1381, … |
| maximal quotient of | #7154, #7172, #7219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 113);