Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C43 · Schur multiplier (H2) = C27×C43 |
| direct product | C43×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C43 | C22 |
| commutator | C2 | C22×C43 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
7878 conjugacy classes of subgroups; only the 4554 normal subgroups are shown.
This group has 7878 conjugacy classes of subgroups — too many to draw. Only its 4554 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2,3,4);(5,6,7,8);(9,10,11,12);(13,14,15,16);(14,16)320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C4×#21503, C42×D42, #72279471, … |
| maximal normal subgroup of | C4×#21503, C42×D42, #72279471, … |
| maximal quotient of | C4×#20512 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420521);