Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7414 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#7414 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1345 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #423 | C23 |
| prime radical | C2×#575 | C4 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#423 ◁ G | |
| LECS | G ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
482 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 482 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2);(4,8,9,13,11,16,10,17)(5,12)(6,14);(3,4)(5,13)(6,9)(7,11)(8,15)(10,14)(12,17)(16,18)50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57424);