Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| 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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
247 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 247 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,11,8)(7,15,18,20)(9,14,10,13)(12,19,16,17);(5,7,11,18)(6,12,8,16)(9,17,10,19)(13,15,14,20);(2,4)(5,8,11,6)(7,16,18,12)(9,14,10,13)(15,17,20,19);(5,9)(6,13)(8,14)(10,11)(15,20)(17,19)Polycyclic presentation:
2^7|a^2=g,b^2=g,c^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,d^b=de,e^b=eg,e^d=eg35 conjugacy classes — view the full character table.
| maximal subgroup of | #19759, #19760, #19767, … |
| maximal normal subgroup of | #19759, #19760, #19767, … |
| maximal quotient of | #19851, #19852, #19853, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2016);