Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25435 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4,6,5)(7,8,14,19)(9,20,10,16)(11,25,28,38)(12,29,13,18)(15,24,31,22)(17,27,33,26)(21,35,23,30)(32,36,34,37);(3,5,6,4)(7,9,13,24,14,10,12,22)(8,15,29,16,19,31,18,20)(11,21,27,37,28,23,26,36)(17,30,38,32,33,35,25,34);(4,5)(7,10)(8,16)(9,14)(11,23)(12,24)(13,22)(15,18)(17,32)(19,20)(21,28)(25,30)(26,37)(27,36)(29,31)(33,34)(35,38);(7,11,13,27,14,28,12,26)(8,17,29,38,19,33,18,25)(9,21,24,37,10,23,22,36)(15,30,16,32,31,35,20,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351486);