Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
474 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 474 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 12 points:
(2,3)(5,6,7,8);(1,2,4,3)(9,10,11,12);(5,6)(7,8)(9,10)(11,12);(1,3,4,2)Polycyclic presentation:
2^7|a^2=f,b^2=eg,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=cg38 conjugacy classes — view the full character table.
| maximal subgroup of | #9397, #9405, #9457, … |
| maximal normal subgroup of | #9397, #9405, #9457, … |
| maximal quotient of | #13906, #13910, #13998, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1273);