Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#50576 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4980 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4,7,9)(5,10,6,8)(12,16)(17,18)(19,20,24,28)(21,22,26,25)(23,27,31,34)(29,30,33,32);(3,5,7,6)(4,8,9,10)(21,26)(22,25)(23,31)(27,34);(5,6)(8,10)(11,12,15,16)(13,17,14,18)(19,21)(20,25)(22,28)(23,29)(24,26)(27,32)(30,34)(31,33);(3,6,7,5)(4,8,9,10)(11,13,15,14)(12,17,16,18)(19,22,24,25)(20,26,28,21)(23,30,31,32)(27,33,34,29);(3,4)(5,10)(6,8)(7,9)(11,14,15,13)(12,18,16,17)(19,23,24,31)(20,27,28,34)(21,29,26,33)(22,30,25,32)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613086);