Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#10610 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #396 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | C22⋊Q8 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,7,10,6,9,5,8)(11,12,16,20)(13,21,22,25)(14,19,15,18)(17,24,26,23);(11,13,14,23,16,22,15,24)(12,17,18,21,20,26,19,25);(3,4,6,9)(5,10,7,8)(12,18)(13,22)(14,15)(17,21)(19,20)(25,26)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 149871);