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×#454 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
364 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 364 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);(4,8)(6,13)(9,17)(10,14)(11,15)(16,18);(3,4)(5,9)(6,14)(7,10)(8,13)(11,16)(12,18)(15,17);(3,5,7,12)(4,9,10,18)(6,15,13,11)(8,16,14,17)32 conjugacy classes — view the full character table.
| maximal subgroup of | #13055, #13057, #13357, … |
| maximal normal subgroup of | #13055, #13057, #13357, … |
| maximal quotient of | C2×#1597, C2×#1615, C2×#1598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1748);