Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C42 |
| direct product | C2×#13797 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(7,8,10,9)(11,12,15,13)(14,17,18,16)(20,23)(21,25);(12,13)(16,17)(19,20,22,23)(21,24,25,26);(3,4,5,6)(7,9,10,8)(11,13,15,12)(14,17,18,16)(19,20)(21,24)(22,23)(25,26);(11,14,15,18)(12,16,13,17)(19,21,22,25)(20,24,23,26)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420645);