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×#10605 |
| 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);(3,4,5,10,7,6,8,9)(11,12,16,18,15,19,14,23)(13,20,24,25,21,26,22,17);(3,5,7,8)(4,9,6,10)(11,13,14,22,15,21,16,24)(12,17,18,20,19,25,23,26);(3,6,7,4)(5,10,8,9)(13,21)(14,16)(18,23)(20,26)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 149867);