Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 1024 |
| 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 | #449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
196 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 196 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:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,19)(12,17)(13,18)(14,16)(15,21)(20,23)(22,24);(2,5)(6,8)(9,11,13,21)(10,15,18,19)(12,20,14,22)(16,23,17,24);(10,16)(12,14)(15,23)(17,18)(19,24)(20,22);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,17,18,16)(11,20,21,22)(15,24,19,23)Polycyclic presentation:
2^7|b^2=g,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,d^c=dg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#3111, #60546, #60559, … |
| maximal normal subgroup of | C2×#3111, #60546, #60559, … |
| maximal quotient of | #20742, #21121, #21122, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1843);