Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) = C212 (order 4096) · Inn(G) = C24 · Out(G) = C28 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C24 |
| commutator | C22 | C23×C4 |
| prime radical | C24 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(6,9)(10,12)(13,14)(15,19)(16,17)(18,20);(1,3,4,2)(14,17)(18,19);(5,6)(7,10)(8,9)(11,12)(13,15,16,20)(14,18,17,19);(5,7,8,11)(6,10,9,12)(13,14)(15,18)(16,17)(19,20)Polycyclic presentation:
2^7|a^2=g,b^2=g,c^2=e,d^2=ef,b^a=be,c^a=cf,d^b=de50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2373, #34149, #34151, … |
| maximal normal subgroup of | C2×#2373, #34149, #34151, … |
| maximal quotient of | #13806, #13815, #13816, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1105);