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×#18421 |
| 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);(3,4,8,13)(5,15,18,31)(6,12,7,22)(9,14,25,30)(10,23,28,16)(11,24,20,21)(17,26,19,33)(27,32,29,34);(3,5,8,18)(4,10,13,28)(6,16,7,23)(9,19,25,17)(11,26,20,33)(12,31,22,15)(14,29,30,27)(21,32,24,34);(3,4)(5,10)(6,20)(7,11)(8,13)(9,14)(12,24)(15,32)(16,33)(17,27)(18,28)(19,29)(21,22)(23,26)(25,30)(31,34);(3,6,8,7)(4,11,13,20)(5,16,18,23)(9,21,25,24)(10,26,28,33)(12,30,22,14)(15,29,31,27)(17,34,19,32)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254208);