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) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #102 |
| commutator | C2×C8 | C2×C4 |
| Frattini, cosocle | C4×C8 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
43 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,12,15,14,18,11,19,13,17,20,23,21,24,22,16);(1,2,4,6)(3,8,7,5)(10,15)(11,20)(12,22)(14,21)(16,18)(17,23)(19,24)Polycyclic presentation:
2^7|a^2=d,b^2=eg,c^2=fg,d^2=efg,f^2=g,b^a=bc,c^a=ce,c^b=cf,d^b=def,f^b=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1365, #11939, #11941, … |
| maximal normal subgroup of | C2×#1365, #11939, #11941, … |
| maximal quotient of | #7121, #7274, #7453, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 97);