Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C2×D8 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
186 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,5)(6,7)(9,10)(11,14)(12,15)(13,16);(1,2,4,5)(3,7,8,6)(9,11,13,12)(10,14,16,15);(10,15)(11,12)(14,16);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^7|b^2=eg,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,c^b=cg,d^b=de,f^c=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | #19787, #19791, #19835, … |
| maximal normal subgroup of | #19787, #19791, #19835, … |
| maximal quotient of | #22523, #22544, #22743, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2099);