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 | C16×D16 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | D8 |
| commutator | C8 | C22×C16 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | C8×D8 |
| prime radical | C2×D16 | C8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C4×C16 ◁ C8×C16 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
175 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 175 conjugacy classes of subgroups — too many to draw. Only its 76 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);(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)176 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C16×D16, #72225391, #72225437, … |
| maximal normal subgroup of | C2×C16×D16, #72225391, #72225437, … |
| maximal quotient of | #72225355, #72225356, C2×C16×D16 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30137);