Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C16 · Schur multiplier (H2) = C2×C8 |
| direct product | C16×M32 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C16 | C22 |
| commutator | C2 | C2×C8×C16 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | C2×C82 |
| prime radical | C23 | C82 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C16 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
264 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16);(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32);(18,26)(20,28)(22,30)(24,32)320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | M322, C16×#504, #72226969, … |
| maximal normal subgroup of | M322, C16×#504, #72226969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29401);