Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#29231 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | #2571 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1034 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);(3,4,9,14)(5,15)(6,20,21,25)(7,10,8,13)(11,16)(12,23,28,22)(17,31)(18,26)(19,24)(27,33)(29,30)(32,34);(4,10,14,13)(5,16,19,18)(6,21)(11,26)(12,20,28,25)(15,24)(17,32,33,30)(22,23);(3,5)(4,11,14,26)(6,17)(7,16)(8,18)(9,19)(10,24,13,15)(12,27,28,31)(20,29,25,34)(21,33)(22,30)(23,32);(3,6,7,22,9,21,8,23)(4,12,13,20,14,28,10,25)(5,17,16,30,19,33,18,32)(11,27,15,29,26,31,24,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375312);