Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
| direct product | C2×#7101 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | D8 |
| commutator | C8 | C22×C16 |
| Frattini, cosocle | C82 | C23 |
| socle, prime radical | C23 | #335 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
194 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 512 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2);(4,8)(5,12)(6,13)(9,14)(10,18)(11,17)(15,16)(19,20,21,24,22,25,27,30,23,26,28,31,29,32,33,34);(3,4,9,18,6,10,5,8,7,11,12,16,13,15,14,17)176 conjugacy classes — view the full character table.
| maximal subgroup of | #72225365, C2×#122946, #72225375, … |
| maximal normal subgroup of | #72225365, C2×#122946, #72225375, … |
| maximal quotient of | #72225362 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30125);