Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) = #72234537 (order 8192) · Inn(G) = C24 · Out(G) = #6342371 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
214 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 214 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)(10,13)(14,15)(17,18,20,22)(19,23,24,21);(2,4)(3,7)(9,10,12,13)(11,15,16,14)(17,18,20,22)(19,21,24,23);(1,3,5,7)(2,6,4,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24);(1,4,5,2)(3,8,7,6)(9,11)(10,14)(12,16)(13,15)(17,19,20,24)(18,21,22,23)Polycyclic presentation:
2^7|a^2=eg,b^2=f,c^2=fg,d^2=ef,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de32 conjugacy classes — view the full character table.
| maximal subgroup of | #7746, #7773, #7779, … |
| maximal normal subgroup of | #7746, #7773, #7779, … |
| maximal quotient of | #14445, #14446, #14471, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1540);