Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) = C22×#72232847 (order 8192) · Inn(G) = C23 · Out(G) = C23×#2846 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C23 |
| commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | #392 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
348 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4)(9,10)(11,14)(12,13)(15,16);(5,6)(7,8)(10,13)(14,16);(2,4)(6,8);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^7|a^2=f,d^2=e,b^a=be,c^a=cf,c^b=cg44 conjugacy classes — view the full character table.
| maximal subgroup of | #9035, #9036, #11691, … |
| maximal normal subgroup of | #9035, #9036, #11691, … |
| maximal quotient of | #13839, #13842, #13847, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1124);