Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10690 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10691 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #401 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C26 | C23 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,11,13,8)(6,9,14,16)(7,10)(12,17)(15,18)(19,20)(21,27)(22,30,32,25)(23,26)(24,29)(28,34,31,33);(3,5)(4,8)(6,12,14,18)(7,13)(9,15,16,17)(10,11)(19,21)(20,24,26,27)(22,31)(23,29)(25,33,30,34)(28,32);(3,6)(4,9,10,16)(5,12)(7,14)(8,15,11,17)(13,18)(19,22)(20,25)(21,28,29,31)(23,32)(24,33,27,34)(26,30)50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151690);