Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18414 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18425 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3536 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3218 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);(3,4,9,6)(5,15,19,29)(7,13,8,21)(10,14,26,22)(11,24,16,17)(12,25,20,23)(18,31,33,27)(28,34,32,30);(3,5,9,19)(4,11,6,16)(7,17,8,24)(10,18,26,33)(12,27,20,31)(13,29,21,15)(14,28,22,32)(23,30,25,34);(3,6,9,4)(5,16,19,11)(7,12,8,20)(10,22,26,14)(13,25,21,23)(15,30,29,34)(17,27,24,31)(18,32,33,28);(3,7,9,8)(4,12,6,20)(5,17,19,24)(10,23,26,25)(11,27,16,31)(13,22,21,14)(15,28,29,32)(18,30,33,34)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254212);