Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13561 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#13565 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | #124 | C24 |
| prime radical | C22×H(5,𝔽2) | C4 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×#124 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1546 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 1546 conjugacy classes of subgroups — too many to draw. Only its 194 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);(3,4,6,11,8,13,20,29)(5,14,16,27,15,30,31,34)(7,12,23,28,22,10,19,26)(9,18,25,21,24,32,33,17);(3,5)(4,9)(6,18)(7,21)(8,15)(10,14)(11,27)(12,30)(13,24)(16,23)(17,22)(19,31)(20,32)(25,28)(26,33)(29,34);(5,15)(9,24)(14,30)(16,31)(17,21)(18,32)(25,33)(27,34)56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#152020, C2×#152681, C2×#152715, … |
| maximal normal subgroup of | C2×#152020, C2×#152681, C2×#152715, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419765);