Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#413 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22⋊C4 |
| commutator | C22 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| socle | C22 | C2×C22⋊C4 |
| prime radical | C22×#41 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
378 conjugacy classes of subgroups; only the 172 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 172 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,7,12,8,11,6,10)(5,13,14,18,15,9,16,17);(3,5)(4,9)(6,14)(7,16)(8,15)(10,17)(11,13)(12,18);(3,6,8,7)(4,10,11,12)(5,14,15,16)(9,17,13,18)44 conjugacy classes — view the full character table.
| maximal subgroup of | #10744, #10746, #10747, … |
| maximal normal subgroup of | #10744, #10746, #10747, … |
| maximal quotient of | C2×#1458, C2×#1459, #17349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1619);