Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C22×#1360 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C23×D8 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1206 conjugacy classes of subgroups; only the 362 normal subgroups are shown.
This group has 1206 conjugacy classes of subgroups — too many to draw. Only its 362 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2);(3,4);(5,6,7,10,8,9,11,12)(14,19,20,27)(15,23)(16,17)(21,24,22,28)(25,26);(6,9)(7,11)(13,14)(15,24)(16,22)(17,21)(18,20)(19,25)(23,27)(26,28)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390233);