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×#554 |
| 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 | |
600 conjugacy classes of subgroups; only the 444 normal subgroups are shown.
This group has 600 conjugacy classes of subgroups — too many to draw. Only its 444 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);(3,4,7,6)(5,9,10,8)(11,12,15,14)(13,17,18,16);(4,6)(8,9)(11,13,15,18)(12,16,14,17);(3,5)(4,8)(6,9)(7,10);(3,6,7,4)(5,9,10,8)(11,14,15,12)(13,16,18,17)50 conjugacy classes — view the full character table.
| maximal subgroup of | #8372, #8380, #8395, … |
| maximal normal subgroup of | #8372, #8380, #8395, … |
| maximal quotient of | C2×#2345, C2×D4×C4⋊C4, C2×#2349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2204);