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 | #127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
175 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 175 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) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,7)(9,10,14,17)(11,15,13,22)(12,19,21,23)(16,20,24,18);(1,2,4,5)(3,7,8,6)(9,11,14,13)(10,15,17,22)(12,20,21,18)(16,19,24,23);(9,12,14,21)(10,16,17,24)(11,18,13,20)(15,23,22,19);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^7|a^2=g,b^2=eg,c^2=g,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 | #19798, #19802, #19846, … |
| maximal normal subgroup of | #19798, #19802, #19846, … |
| maximal quotient of | #22525, #22546, #22743, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2107);