Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10370 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#10689 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #400 |
| prime radical | C24 | C22⋊Q8 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
572 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);(3,4,6,7)(5,8,10,9)(11,12,15,14,16,18,20,24)(13,19,21,26,22,23,25,17);(3,5,6,10)(4,8,7,9)(11,13,20,21,16,22,15,25)(12,17,24,19,18,26,14,23);(4,7)(5,9)(8,10)(11,14,15,18,16,24,20,12)(13,19,21,26,22,23,25,17)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151274);