Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C29 |
| direct product | C2×#17787 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator | C4 | C24×C8 |
| Frattini, cosocle | C42 | C25 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C25 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2318 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);(3,4,5,7,6,8,9,10)(11,12)(13,20)(14,18)(15,24)(16,19)(17,23)(21,26)(22,25);(11,13,15,23,16,21,14,22)(12,17,18,20,19,25,24,26);(13,21)(17,25)(20,26)(22,23);(11,14,16,15)(12,18,19,24)(13,22,21,23)(17,20,25,26)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251502);