Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) = C24×D42 (order 1024) · Inn(G) = #121 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C2×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
141 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 141 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(7,15)(8,11)(9,13)(10,14)(12,18)(16,19)(17,20);(6,11)(7,16)(8,9)(12,15)(13,14)(19,20);(2,4)(5,7,8,17,10,16,9,18)(6,12,13,19,14,20,11,15)Polycyclic presentation:
2^7|c^2=d,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^c=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2051, #23530, #23531, … |
| maximal normal subgroup of | C2×#2051, #23530, #23531, … |
| maximal quotient of | #10456, #10457, #10460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 783);