Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#7852 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | C23 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C24 | C2×C42 |
| prime radical | C25 | C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2366 conjugacy classes of subgroups; only the 834 normal subgroups are shown.
This group has 2366 conjugacy classes of subgroups — too many to draw. Only its 834 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) = 256 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(7,8,9,12,10,11,13,14);(4,6)(15,16,17,20,18,19,21,22);(8,11)(12,14)(16,19)(20,22)176 conjugacy classes — view the full character table.
| maximal subgroup of | #72209971, C2×#7852⋊C3, C2×#233521, … |
| maximal normal subgroup of | #72209971, C2×#7852⋊C3, C2×#233521, … |
| maximal quotient of | C2×#105250, C2×#94876, C2×#94877 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88050);