Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C16 · Schur multiplier (H2) = C29 |
| direct product | C16×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C24 |
| commutator | C2 | C24×C16 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C22 | C24×C8 |
| prime radical | C2×2−1+4 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1444 conjugacy classes of subgroups; only the 1304 normal subgroups are shown.
This group has 1444 conjugacy classes of subgroups — too many to draw. Only its 1304 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 — 5 generators on 32 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16);(18,22)(20,28)(23,25)(26,32);(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30);(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)272 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529978);