Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#29232 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | #2572 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,7,11,8,12,6,9)(5,13,14,18,15,10,16,17);(4,9,12,11)(5,14,15,16)(10,13)(17,18);(3,5)(4,10)(6,14)(7,16)(8,15)(9,17)(11,18)(12,13);(4,9,12,11)(10,17,13,18)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6368587);