Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#30869 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2761 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(7,8,13,17)(9,18,15,20)(10,19,14,11)(12,22,21,16);(3,4)(5,6)(7,9,13,15)(8,14,17,10)(11,21,19,12)(16,20,22,18);(4,6)(7,10)(8,15)(9,17)(11,16)(12,20)(13,14)(18,21)(19,22);(7,11,12,9,13,19,21,15)(8,10,16,20,17,14,22,18)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409342);