Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) = #72231459 (order 2048) · Inn(G) = D8 · Out(G) = C4×#120 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D8 |
| commutator | C8 | C42 |
| Frattini, cosocle | C22×C8 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(10,15)(11,19)(12,21)(13,20)(16,17)(18,24)(22,23);(2,4)(9,10,12,16,13,17,11,15,14,18,19,23,20,22,21,24)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=fg,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 | #7418, #7419, C2×#1380, … |
| maximal normal subgroup of | #7418, #7419, C2×#1380, … |
| maximal quotient of | #7154, #7171, #7217, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 112);