Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#388 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C23 |
| commutator | C22 | C23×C4 |
| Frattini, cosocle | C23 | C24 |
| prime radical | C25 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
464 conjugacy classes of subgroups; only the 196 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 196 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2);(4,6)(8,9)(11,12,14,13);(7,8,10,9)(11,13,14,12);(3,4)(5,6)(7,9,10,8)56 conjugacy classes — view the full character table.
| maximal subgroup of | #8155, #8561, #9080, … |
| maximal normal subgroup of | #8155, #8561, #9080, … |
| maximal quotient of | C2×#1434, C2×#1436, C2×#1437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1021);