Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12364 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#12364 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1629 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#3291 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
963 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 963 conjugacy classes of subgroups — too many to draw. Only its 154 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 — 4 generators on 18 points:
(1,2);(4,8,10,18)(5,12)(6,15)(9,16,11,13)(14,17);(5,12)(6,15)(8,16)(9,11)(13,18)(14,17);(3,4)(5,13)(6,9)(7,10)(8,17)(11,15)(12,18)(14,16)68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214877, D3×#12364, C2×#489347, … |
| maximal normal subgroup of | #72214877, C2×#489347, #72215869, … |
| maximal quotient of | #72214877, D3×#12364, #72219624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396121);