Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C22×D4×#120 (order 1024) · Inn(G) = C2×D4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C42 | C23 |
| socle | C22 | C2×C4⋊C4 |
| prime radical | C23 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
79 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,6,8)(3,4,7,5)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24);(1,3,5,2,6,7,4,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(1,4,6,5)(2,7,8,3)(11,17)(14,20)(18,23)(21,24)Polycyclic presentation:
2^7|a^2=e,b^2=d,c^2=f,d^2=f,e^2=g,b^a=bd,d^a=df,c^b=cfg38 conjugacy classes — view the full character table.
| maximal subgroup of | #8898, #8899, #8900, … |
| maximal normal subgroup of | #8898, #8899, #8900, … |
| maximal quotient of | #7880, #8279, #8600, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 306);