Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13609 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#13609 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #127 | C24 |
| prime radical | C2×#3220 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
910 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 910 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,14,16,9)(6,10)(7,12,22,11)(8,13)(15,29,30,23)(17,24,21,25)(18,31,32,34)(19,27)(20,26,33,28);(3,5)(4,9)(6,18)(7,21)(8,17)(10,23)(11,24)(12,14)(13,25)(15,19)(16,22)(20,30)(26,29)(27,34)(28,31)(32,33);(3,6)(4,10)(5,15)(7,20)(8,19)(9,23)(11,28)(12,26)(13,27)(14,29)(16,30)(17,18)(21,32)(22,33)(24,31)(25,34)50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#512663, C2×#152133, C2×#152136, … |
| maximal normal subgroup of | C2×#512663, C2×#152133, C2×#152136, … |
| maximal quotient of | #72285394, #72293103, #72293104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419856);