Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #27987 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#27987 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2504 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,8,29,10,15,9,14)(5,16,18,11,19,22,30,31)(6,20,24,32,25,34,23,12)(7,17,26,13,28,21,27,33);(3,5,8,18,10,19,9,30)(4,11,14,16,15,31,29,22)(6,21,23,13,25,17,24,33)(7,12,26,20,28,32,27,34);(3,6)(4,12,15,32)(5,17,19,21)(7,22)(8,23)(9,24)(10,25)(11,26)(13,18,33,30)(14,20,29,34)(16,28)(27,31);(3,7,8,26,10,28,9,27)(4,13,14,17,15,33,29,21)(5,12,18,20,19,32,30,34)(6,22,23,11,25,16,24,31)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365726);