Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13569 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13569 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | #124 | C24 |
| prime radical | C22×#454 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×#124 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1522 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 1522 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)(5,13,16,31)(6,18,20,30)(7,22,24,11)(8,12)(9,15,26,33)(10,21)(14,28,17,32)(19,27,34,25)(23,29);(3,5)(4,9)(6,19)(7,16)(8,14)(10,28)(11,26)(12,25)(13,29)(15,20)(17,24)(18,32)(21,33)(22,27)(23,34)(30,31);(5,14)(9,25)(13,28)(15,19)(16,17)(26,27)(31,32)(33,34)56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419772);