Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C22×C4×#120 (order 512) · Inn(G) = C23 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C23 |
| commutator, socle, prime radical | C22 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
63 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10,12,16,13,11,18,19,14,17,21,24,22,20,15,23);(1,3,5,4)(2,6,7,8);(1,4,5,3)(2,6,7,8)(9,11,12,19,13,17,18,24,14,20,21,23,22,10,15,16)Polycyclic presentation:
2^7|a^2=e,b^2=d,c^2=de,e^2=f,f^2=g,b^a=bd,c^a=cg56 conjugacy classes — view the full character table.
| maximal subgroup of | #7393, #12049, #12051, … |
| maximal normal subgroup of | #7393, #12049, #12051, … |
| maximal quotient of | #12662, #12663, #13104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 915);