Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#59720 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(4,6)(8,9)(11,12,16,14)(13,18,15,17)(20,23)(21,25);(3,4,7,6)(5,9,10,8)(11,13,16,15)(12,17,14,18)(20,23)(21,25);(11,12,16,14)(13,17,15,18)(19,20)(21,24)(22,23)(25,26);(3,5,7,10)(4,8,6,9)(11,14,16,12)(13,18,15,17)(19,21,22,25)(20,24,23,26);(3,6,7,4)(5,9,10,8)(11,15,16,13)(12,18,14,17)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7573747);