Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
| direct product | C2×#32901 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C22×#585 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
17280 conjugacy classes of subgroups; only the 954 normal subgroups are shown.
This group has 17280 conjugacy classes of subgroups — too many to draw. Only its 954 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(3,4)(5,6)(7,8,12,11)(9,13,17,16)(10,14,19,18)(15,21,22,20);(7,9)(8,13)(10,15)(11,16)(12,17)(14,20)(18,21)(19,22);(7,10)(8,14)(9,15)(11,18)(12,19)(13,20)(16,21)(17,22);(4,6)(7,11,12,8)(9,16,17,13)(10,18,19,14)(15,21,22,20)98 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6628891, C2×#6628900, #72251698, … |
| maximal normal subgroup of | C2×#6628891, C2×#6628900, #72251698, … |
| maximal quotient of | C2×#346481, C2×#346476, C2×#347049, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536086);