Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C2×#72218460 (order 2048) · Inn(G) = C23 · Out(G) = C23×#120 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C23 |
| commutator, socle | C22 | C22×C8 |
| prime radical | C24 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,7,11,8,12,13,18,9,10,14,16,15,17,19,20);(2,4);(6,10)(11,16)(12,17)(18,20)Polycyclic presentation:
2^7|a^2=e,e^2=f,f^2=g,b^a=bd,c^a=cg56 conjugacy classes — view the full character table.
| Frattini subgroup of | #93675, #93676, #93685, … |
| maximal subgroup of | #7184, #7185, #7200, … |
| maximal normal subgroup of | #7184, #7185, #7200, … |
| maximal quotient of | #11418, #11440, #11460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 844);