Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#24149 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12)(13,18)(14,16)(15,19)(17,24)(20,25)(21,26)(22,23);(4,7)(8,9)(11,13)(12,18)(14,23)(15,21)(16,22)(17,25)(19,26)(20,24);(11,14,16,12)(13,20,22,26)(15,24,17,19)(18,25,23,21);(3,5,6,10)(4,8,7,9)(11,15,16,17)(12,19,14,24)(13,21,22,25)(18,26,23,20)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355855);