Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C22 |
| commutator | C2 | C2×C4×C8 |
| Frattini, cosocle | C42 | C23 |
| socle | C22 | C2×C42 |
| prime radical | C23 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24);(1,2,3,5,4,6,7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(11,17)(14,20)(18,23)(21,24)Polycyclic presentation:
2^7|a^2=d,b^2=e,d^2=f,e^2=g,c^b=cf80 conjugacy classes — view the full character table.
| maximal subgroup of | #7140, #7141, #7153, … |
| maximal normal subgroup of | #7140, #7141, #7153, … |
| maximal quotient of | #7841, #7849, #8133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 182);