Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| 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 | #469 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
177 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 177 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,7)(9,10)(11,18)(12,16)(13,15)(14,19)(17,21)(20,23)(22,24);(1,2)(3,6)(4,5)(7,8)(9,11,12,19)(10,14,16,18)(13,20,21,24)(15,22,17,23);(2,5)(6,8)(10,15)(13,21)(14,22)(16,17)(18,23)(20,24);(1,3,4,7)(2,6,5,8)(10,16)(14,18)(15,17)(22,23)Polycyclic presentation:
2^7|b^2=g,d^2=eg,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,c^b=ceg,f^c=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | #12175, #12309, C2×#3256, … |
| maximal normal subgroup of | #12175, #12309, C2×#3256, … |
| maximal quotient of | #21895, #21965, #22120, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1988);