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 | #535 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
245 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 245 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)(7,13)(8,15)(10,17)(12,19)(14,16)(18,20);(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(13,17)(14,18)(15,19)(16,20);(5,7,9,14)(6,10,11,18)(8,16,15,13)(12,20,19,17);(6,11)(10,18)(12,19)(17,20)Polycyclic presentation:
2^7|b^2=e,c^2=g,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 | #24410, #25181, #25184, … |
| maximal normal subgroup of | #24410, #25181, #25184, … |
| maximal quotient of | #19396, #19397, #19398, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1775);