Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#464 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
250 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(4,6)(7,8)(9,15)(10,12)(11,19)(13,17)(14,18)(16,22)(20,21);(3,4,5,6)(8,12)(13,21)(14,19)(15,22);(7,9,10,16)(8,13,12,21)(11,20,18,17)(14,22,19,15)38 conjugacy classes — view the full character table.
| maximal subgroup of | #9244, C4×#464, #21508, … |
| maximal normal subgroup of | #9244, C4×#464, #21508, … |
| maximal quotient of | C2×#1678, C2×#1680, C2×#1682, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1766);