Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18524 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#18547 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3500 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);(4,7)(5,9)(8,10)(11,12)(13,20)(14,18)(15,23)(16,19)(17,21)(22,25)(24,26);(5,9)(7,10)(11,13,16,22)(12,17,19,26)(14,21,15,24)(18,25,23,20);(3,4,6,8)(5,10,9,7)(11,14,16,15)(12,18,19,23)(13,21,22,24)(17,25,26,20);(12,19)(17,26)(18,23)(20,25)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255291);