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) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C24 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
208 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,18,21,23)(19,20,24,22);(2,4)(6,7)(9,10,12,14)(11,15,16,13)(17,19,21,24)(18,22,23,20);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16);(1,4,5,2)(3,7,8,6)(9,11,12,16)(10,13,14,15)(17,20,21,22)(18,19,23,24)Polycyclic presentation:
2^7|a^2=eg,b^2=ef,d^2=e,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 | #12178, #12254, #36729, … |
| maximal normal subgroup of | #12178, #12254, #36729, … |
| maximal quotient of | #14370, #14402, #14459, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1474);