Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×#72213548 (order 2048) · Inn(G) = D4 · Out(G) = C23×H(5,𝔽2) |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | D4 |
| commutator | C4 | C22×C8 |
| Frattini, cosocle | C42 | C23 |
| socle | C22 | C2×C4⋊C4 |
| prime radical | C23 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,5,4,6,7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,23)(21,24);(9,11,13,20,14,21,12,19)(10,15,16,18,17,23,22,24);(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,18,23,24)Polycyclic presentation:
2^7|a^2=e,b^2=d,c^2=f,d^2=f,e^2=g,b^a=bd,d^a=df56 conjugacy classes — view the full character table.
| maximal subgroup of | #7155, #7157, #7158, … |
| maximal normal subgroup of | #7155, #7157, #7158, … |
| maximal quotient of | #7872, C4×#334, C4×#335, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 296);