Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) = #72234537 (order 8192) · Inn(G) = C24 · Out(G) = #6342371 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C24 |
| commutator | C22 | C23×C4 |
| prime radical | C2×#121 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
348 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 148 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:
(2,4)(5,6,8,7)(9,10)(11,13)(12,15)(14,16);(5,7,8,6)(10,11)(14,15);(1,2,3,4)(9,11,13,10)(12,15,16,14);(9,12)(10,14)(11,15)(13,16)Polycyclic presentation:
2^7|a^2=g,b^2=g,c^2=f,b^a=be,c^a=cf,d^a=de,c^b=ce44 conjugacy classes — view the full character table.
| maximal subgroup of | #8866, #8869, #8870, … |
| maximal normal subgroup of | #8866, #8869, #8870, … |
| maximal quotient of | #13778, #13784, #13789, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1057);