Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19072 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19081 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,9)(5,13)(6,7)(11,12)(14,15)(16,17)(19,20,25,22)(21,29,32,34)(23,28,24,27)(26,33,30,31);(3,4)(5,10)(6,11)(7,9)(8,12)(13,17)(14,16)(15,18)(19,21)(20,26)(22,30)(23,31)(24,33)(25,32)(27,29)(28,34);(3,5,8,15)(4,10,12,18)(6,13,7,14)(9,16,11,17)(19,22,25,20)(21,30,32,26)(23,27,24,28)(29,33,34,31);(3,6,8,7)(4,11,12,9)(5,14,15,13)(10,16,18,17)(19,23,25,24)(20,27,22,28)(21,31,32,33)(26,29,30,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258690);