Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#8735 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #381 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,5,7,6,8,9,10)(11,12)(13,19,21,16)(14,22)(15,18)(17,23)(20,24,26,25);(11,13,15,21)(12,16,18,19)(14,20,23,26)(17,24,22,25);(11,14)(12,17,18,22)(13,20)(15,23)(16,24,19,25)(21,26)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97207);