Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| socle | C22 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
260 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 104 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)(7,14)(8,12)(9,18)(10,13)(11,16)(15,19)(17,20);(5,7,9,16,10,17,8,15)(6,11,12,14,13,19,18,20);(2,4);(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,14,19,20)Polycyclic presentation:
2^7|b^2=e,d^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg44 conjugacy classes — view the full character table.
| maximal subgroup of | #11703, #11704, #11709, … |
| maximal normal subgroup of | #11703, #11704, #11709, … |
| maximal quotient of | #19101, #19105, #19110, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1782);