Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | Aut(G) = C2×D4×#442 (order 1024) · Inn(G) = #102 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #102 |
| commutator | C2×C8 | C2×C4 |
| Frattini, cosocle | C4×C8 | C22 |
| prime radical | #493 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
59 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10)(11,14)(12,21)(13,17)(15,18)(16,19)(20,23)(22,24);(2,5)(3,7)(10,14)(11,18)(12,19)(15,16)(17,23)(20,24)(21,22)Polycyclic presentation:
2^7|a^2=d,c^2=fg,d^2=efg,e^2=g,f^2=g,b^a=bc,c^a=ce,e^a=eg,f^a=fg,c^b=cf,d^b=defg,e^b=eg,f^b=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1361, #11929, #11930, … |
| maximal normal subgroup of | C2×#1361, #11929, #11930, … |
| maximal quotient of | #7094, #7096, #7097, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 93);