Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) order 4096 |
| 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 | #532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
185 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 185 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)(3,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24);(1,2,4,5)(3,7,8,6)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22);(1,3,4,8)(2,6,5,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23);(1,2,4,5)(3,6,8,7)(10,16)(14,17)(15,24)(22,23)Polycyclic presentation:
2^7|b^2=e,c^2=e,d^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,c^b=ce,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #9279, #9284, #18764, … |
| maximal normal subgroup of | #9279, #9284, #18764, … |
| maximal quotient of | #21835, #21863, #21984, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1960);