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) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C23 |
| commutator | C23 | C22×C4 |
| prime radical | C25 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
390 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 390 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 — 3 generators on 16 points:
(6,8)(9,10,11,12)(14,16);(2,4)(6,8)(13,14,15,16);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)Polycyclic presentation:
2^7|a^2=g,b^2=d,b^a=bd,c^a=ce,c^b=cf44 conjugacy classes — view the full character table.
| maximal subgroup of | #7250, #8403, #8405, … |
| maximal normal subgroup of | #7250, #8403, #8405, … |
| maximal quotient of | #7555, #7556, #7557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 171);