Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C211×C4 |
| direct product | C2×#37216 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6042 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)(5,9)(11,12,17,15)(13,18,16,14)(19,20,22,24)(21,23,25,26);(3,4,7,6)(5,10,9,8)(11,13,17,16)(12,18,15,14);(3,4,7,6)(5,8,9,10)(11,14)(12,16)(13,15)(17,18)(19,20)(21,23)(22,24)(25,26);(3,5,7,9)(4,8,6,10)(11,15,17,12)(13,18,16,14);(3,6,7,4)(5,10,9,8)(11,16,17,13)(12,18,15,14)(19,21,22,25)(20,23,24,26)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541785);