Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18523 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator, Frattini, cosocle | C42 | C25 |
| socle | C23 | C2×#120 |
| prime radical | C2×#3475 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,8,11)(5,12,14,17)(6,10,7,15)(9,16,18,13)(19,20,25,29)(21,30,22,33)(23,27,24,28)(26,31,34,32);(3,5,8,14)(4,9,11,18)(6,13,7,16)(10,17,15,12)(19,21,25,22)(20,26,29,34)(23,31,24,32)(27,30,28,33);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,17,18,12)(19,22,25,21)(20,26,29,34)(23,32,24,31)(27,30,28,33);(19,23,25,24)(20,27,29,28)(21,31,22,32)(26,30,34,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255262);