Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#35775 |
| 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 | |
5854 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);(3,4)(5,10)(6,8)(7,9)(11,12,16,18)(13,15,17,14)(20,23)(24,26);(3,5,7,6)(4,8,9,10)(11,13,16,17)(12,14,18,15)(19,20,22,23)(21,24,25,26);(3,4)(5,10)(6,8)(7,9)(11,12)(13,14)(15,17)(16,18);(3,6,7,5)(4,8,9,10)(11,14,16,15)(12,13,18,17);(3,4)(5,8)(6,10)(7,9)(11,15,16,14)(12,17,18,13)(19,21,22,25)(20,24,23,26)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536540);