Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#18997 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C22×H(3,𝔽4) | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,8,14)(5,9,16,24)(6,10,18,27)(7,11,20,28)(12,29,30,15)(13,21,25,19)(17,33,32,22)(23,31,26,34);(4,9)(5,15)(6,18)(10,22)(11,12)(13,26)(14,24)(16,29)(17,34)(19,21)(23,25)(27,33)(28,30)(31,32);(4,10)(5,16)(6,19)(9,22)(11,25)(12,23)(13,28)(14,27)(15,29)(17,31)(18,21)(24,33)(26,30)(32,34);(4,11)(9,12)(10,25)(13,27)(14,28)(22,23)(24,30)(26,33)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258574);