Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#397 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C23 |
| commutator, Frattini, cosocle | C23 | C24 |
| prime radical | C25 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
410 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 148 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 — 4 generators on 18 points:
(1,2);(7,8,9,10)(12,15)(16,17);(4,6)(11,12)(13,17)(14,15)(16,18);(3,4)(5,6)(7,8)(9,10)(11,13,14,18)(12,16,15,17)44 conjugacy classes — view the full character table.
| maximal subgroup of | #9025, #9028, #9918, … |
| maximal normal subgroup of | #9025, #9028, #9918, … |
| maximal quotient of | C2×#1439, C2×#1440, C2×#1442, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1122);