Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#41742 |
| 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 | |
5114 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);(5,9)(8,10)(11,12)(13,15)(14,17)(16,18)(19,20,24,25)(21,27,30,26)(22,31,33,23)(28,34,32,29);(3,4,7,6)(5,8,9,10)(12,13)(16,17)(20,25)(23,31)(26,27)(29,34);(3,5,7,9)(4,8,6,10)(11,13,15,12)(14,17,18,16)(19,21,24,30)(20,26,25,27)(22,32,33,28)(23,29,31,34);(11,14)(12,16)(13,17)(15,18)(19,22)(20,23)(21,28)(24,33)(25,31)(26,34)(27,29)(30,32);(3,6,7,4)(5,10,9,8)(19,23)(20,22)(21,29)(24,31)(25,33)(26,32)(27,28)(30,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7546860);