Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = #72232064 (order 2048) · Inn(G) = C2×D4 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C4 | C22×C8 |
| Frattini, cosocle | C42 | C23 |
| socle | C22 | C2×C22⋊C4 |
| prime radical | C23 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
124 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);(9,11,14,21)(10,15,17,23)(12,20,13,19)(16,24,22,18);(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=f,c^2=f,d^2=f,e^2=g,b^a=bd,d^a=df,c^b=cf,d^b=df44 conjugacy classes — view the full character table.
| maximal subgroup of | #8323, #8324, #8325, … |
| maximal normal subgroup of | #8323, #8324, #8325, … |
| maximal quotient of | #7573, #7575, #7581, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 211);