Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
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 | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,15,17,16,18,19,20);(1,2,4,3)(6,9)(10,12);(1,3,4,2)(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^7|a^2=f,b^2=e,c^2=d,f^2=g,b^a=bd,c^a=ce56 conjugacy classes — view the full character table.
| Frattini subgroup of | #93689, #93690, #93698, … |
| maximal subgroup of | #7018, #7022, #7030, … |
| maximal normal subgroup of | #7018, #7022, #7030, … |
| maximal quotient of | #8780, #8781, #8784, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 563);