Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#538 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C24 |
| commutator, Frattini, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
668 conjugacy classes of subgroups; only the 396 normal subgroups are shown.
This group has 668 conjugacy classes of subgroups — too many to draw. Only its 396 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 — 5 generators on 18 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(12,15)(13,17);(4,7)(8,9)(11,12,14,15)(13,16,17,18);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,18);(3,5)(4,8)(6,10)(7,9)(11,13)(12,16)(14,17)(15,18)44 conjugacy classes — view the full character table.
| maximal subgroup of | #8408, #17386, #17405, … |
| maximal normal subgroup of | #8408, #17386, #17405, … |
| maximal quotient of | C2×#2312, C2×#2318, C2×#2321, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2182);