Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) = #72234537 (order 8192) · Inn(G) = C24 · Out(G) = #6342371 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | #392 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
236 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 236 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)(6,8)(9,10,12,13)(11,14,16,15)(17,18)(19,22)(20,21)(23,24);(3,7)(6,8)(10,13)(14,15)(18,21)(22,23);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,19)(18,22)(20,24)(21,23);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,23)(20,21)(22,24)Polycyclic presentation:
2^7|a^2=g,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de26 conjugacy classes — view the full character table.
| maximal subgroup of | #12136, #12230, C2×#2824, … |
| maximal normal subgroup of | #12136, #12230, C2×#2824, … |
| maximal quotient of | #14621, #14622, #14635, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1556);