Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18414 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18420 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3608 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);(4,8)(5,6)(9,10)(11,12)(13,20)(14,18)(15,23)(16,19)(17,21)(22,25)(24,26);(3,4,7,9)(5,8,6,10)(11,13,16,22)(12,17,19,26)(14,21,15,24)(18,25,23,20);(5,6)(8,10)(11,12)(13,17)(14,23)(15,18)(16,19)(20,21)(22,26)(24,25);(3,5,7,6)(4,8,9,10)(11,14,16,15)(12,18,19,23)(13,21,22,24)(17,25,26,20)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254207);