Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D82 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19971 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(11,12,17,14)(13,21,24,26)(15,20,16,19)(18,25,22,23);(11,13)(12,18)(14,22)(15,23)(16,25)(17,24)(19,21)(20,26);(4,7)(5,10)(8,9)(11,14,17,12)(13,22,24,18)(15,20,16,19)(21,23,26,25);(11,15)(12,19)(14,20)(16,17)(21,26)(23,25)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265556);