Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#24772 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(4,5)(7,8,11,16,13,18,12,17)(9,14,20,29,21,30,27,34)(10,22,24,33,26,32,25,15)(19,31,23,37,35,38,36,28);(3,4,6,5)(7,9)(8,14)(10,23)(11,20)(12,27)(13,21)(15,31)(16,29)(17,34)(18,30)(19,25)(22,37)(24,35)(26,36)(28,32)(33,38);(7,10,13,26)(8,15,18,33)(9,19,21,35)(11,25,12,24)(14,28,30,37)(16,32,17,22)(20,36,27,23)(29,38,34,31);(3,5,6,4)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350220);