Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) order 4096 |
| 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 | C2×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
247 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 247 conjugacy classes of subgroups — too many to draw. Only its 100 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:
(2,4)(5,6,9,13)(7,10,14,18)(8,11,16,12)(15,17,20,19);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,17)(12,19)(13,18)(16,20);(6,11)(8,16)(10,17)(12,13)(15,20)(18,19);(7,14)(10,18)(15,20)(17,19)Polycyclic presentation:
2^7|a^2=g,b^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=dg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #24409, #25182, #25187, … |
| maximal normal subgroup of | #24409, #25182, #25187, … |
| maximal quotient of | #19390, #19391, #19392, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1774);