Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C43 |
| direct product | C4×#2867 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2867 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
7878 conjugacy classes of subgroups; only the 1706 normal subgroups are shown.
This group has 7878 conjugacy classes of subgroups — too many to draw. Only its 1706 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);(6,8)(10,12)(14,16);(5,6,7,8);(9,10,11,12);(13,14,15,16)176 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D4×#127, #72283289 |
| maximal normal subgroup of | C4×D4×#127, #72283289 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 514839);