Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | #525 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
248 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 102 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 — 4 generators on 20 points:
(1,2)(3,4)(5,6,9,12)(7,13,14,20)(8,16,17,11)(10,18,19,15);(5,7,8,15,9,14,17,18)(6,10,11,13,12,19,16,20);(2,4);(7,14)(10,19)(13,20)(15,18)Polycyclic presentation:
2^7|a^2=g,b^2=e,e^2=g,b^a=be,c^a=cf,e^a=eg,d^b=dg32 conjugacy classes — view the full character table.
| maximal subgroup of | #24413, #24415, #24417, … |
| maximal normal subgroup of | #24413, #24415, #24417, … |
| maximal quotient of | #19163, #19165, #19167, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1788);