Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C23 |
| commutator | C22 | C23×C4 |
| Frattini, socle, cosocle | C23 | C24 |
| prime radical | C24 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20);(6,9)(10,12)(13,15,16,19)(14,17,18,20);(14,18)(17,20);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^7|a^2=g,b^2=f,d^2=e,b^a=be,c^a=cf56 conjugacy classes — view the full character table.
| maximal subgroup of | #7857, #7858, #7894, … |
| maximal normal subgroup of | #7857, #7858, #7894, … |
| maximal quotient of | #13753, #13755, #13757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1015);