Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23906 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#24078 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3493 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4)(5,10)(6,8)(7,9)(11,12)(13,18)(14,23)(15,19)(16,24)(17,20)(21,25)(22,26);(3,5,7,6)(4,8,9,10)(11,13)(12,18)(14,16)(15,21)(17,22)(19,25)(20,26)(23,24);(3,6,7,5)(4,8,9,10)(11,14)(12,18)(13,15)(16,22)(17,21)(19,25)(20,26)(23,24);(4,9)(8,10)(11,15,17,16)(12,19,20,24)(13,21,22,14)(18,25,26,23)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360127);