Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25406 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2656 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)(7,8,11,15,12,16,25,32)(9,13,18,27,19,28,36,38)(10,20,22,30,23,31,24,14)(17,33,34,29,21,37,35,26);(3,4,6,5)(7,9,11,18,12,19,25,36)(8,13,15,27,16,28,32,38)(10,21,22,35,23,17,24,34)(14,29,20,37,30,26,31,33);(7,10,12,23)(8,14,16,30)(9,17,19,21)(11,24,25,22)(13,26,28,29)(15,31,32,20)(18,35,36,34)(27,37,38,33);(3,5,6,4)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351511);