Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C26×D4 (order 512) · Inn(G) = C2×D4 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C22 | C4×D4 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
135 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 135 conjugacy classes of subgroups — too many to draw. Only its 56 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)(5,6,10,13,11,8,9,14)(7,15,17,12,19,20,18,16);(2,4)(5,7,10,17,11,19,9,18)(6,12,13,20,8,16,14,15);(5,8)(6,11)(7,16)(9,13)(10,14)(12,19)(15,17)(18,20)Polycyclic presentation:
2^7|a^2=f,b^2=f,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,e^b=eg,d^c=dg,e^c=eg38 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1866, #20884, #20885, … |
| maximal normal subgroup of | C2×#1866, #20884, #20885, … |
| maximal quotient of | #8843, #8844, #8845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 598);