Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#556 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C24 |
| commutator, Frattini, cosocle | C22 | C25 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
556 conjugacy classes of subgroups; only the 404 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 404 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) = 64 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(13,17)(15,18);(3,5,8,6)(4,9,10,7)(11,12,14,16)(13,15,17,18);(3,6,8,5)(4,9,10,7)(11,13)(12,15)(14,17)(16,18);(3,7,8,9)(4,5,10,6)50 conjugacy classes — view the full character table.
| maximal subgroup of | #8365, #8368, #8389, … |
| maximal normal subgroup of | #8365, #8368, #8389, … |
| maximal quotient of | C2×#2347, C2×#2354, C2×#2355, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2206);