Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#10385 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #393 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,6,9)(5,11)(7,10,15,17)(8,16)(12,13)(14,18)(19,20)(21,24)(22,29,31,25)(23,26)(27,32,30,34)(28,33);(3,4)(5,12,14,8)(6,9)(7,10)(11,13,18,16)(15,17)(19,21,23,28)(20,24,26,33)(22,30)(25,34)(27,31)(29,32);(3,5)(4,8)(6,14)(7,16)(9,12)(10,18)(11,17)(13,15)(19,22,23,31)(20,25,26,29)(21,27,28,30)(24,32,33,34)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145569);