Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C23 |
| commutator | C22 | C22×C8 |
| socle, prime radical | C23 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
120 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)(11,13)(12,14)(15,16)(17,18,19,21,20,22,23,24);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16);(1,4,5,3)(2,6,7,8)(10,14)(13,16)Polycyclic presentation:
2^7|a^2=f,b^2=de,c^2=d,f^2=g,b^a=bd,c^a=ce56 conjugacy classes — view the full character table.
| Fitting subgroup of | #1839⋊C3 |
| maximal subgroup of | #7019, #7023, #7041, … |
| maximal normal subgroup of | #7019, #7023, #7041, … |
| maximal quotient of | #8800, #8801, #10795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 571);