Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#20857 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4586 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)(9,17)(10,14)(11,23)(12,15)(13,19)(16,26)(18,34)(20,31)(21,32)(22,29)(24,28)(25,30)(27,35)(33,36)(37,38);(3,5,6,4)(7,9)(8,13)(10,18)(11,24)(12,20)(14,27)(15,28)(16,31)(17,23)(19,26)(21,37)(22,36)(25,35)(29,38)(30,33)(32,34);(7,10,12,22)(8,14,15,29)(9,18,20,36)(11,25,26,21)(13,27,28,38)(16,32,23,30)(17,33,31,34)(19,37,24,35);(8,15)(13,28)(14,29)(16,23)(17,31)(27,38)(30,32)(33,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278481);