Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C22×#2238 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #481 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×#469 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1778 conjugacy classes of subgroups; only the 578 normal subgroups are shown.
This group has 1778 conjugacy classes of subgroups — too many to draw. Only its 578 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(6,10)(7,13)(8,16)(11,18)(12,15)(14,20)(17,19);(7,14)(10,18)(13,20)(15,19);(5,6,8,17,9,11,16,12)(7,15,13,18,14,19,20,10)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#512308, C22×#33688, C2×#512299, … |
| maximal normal subgroup of | C2×#512308, C22×#33688, C2×#512299, … |
| maximal quotient of | C2×#512299 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530252);