Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) = #72252600 (order 4096) · Inn(G) = #120 · Out(G) = C2×D42 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#456 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C2×2−1+4 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
354 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 354 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) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,8,6)(5,11,14,18)(7,15,17,10)(9,16,12,13);(3,5)(4,9)(6,12)(7,16)(8,14)(10,18)(11,15)(13,17);(3,6,8,4)(5,12,14,9)(7,10,17,15)(11,16,18,13)32 conjugacy classes — view the full character table.
| maximal subgroup of | #13056, #13058, #13358, … |
| maximal normal subgroup of | #13056, #13058, #13358, … |
| maximal quotient of | C2×#1603, C2×#1609, C2×#1604, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1749);