Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
| direct product | C2×#30556 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2617 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12)(13,16)(14,25)(15,31)(17,24)(18,21)(19,29)(20,34)(22,30)(23,33)(26,28)(27,37)(32,36)(35,40)(38,42)(39,41);(3,5,6,10)(4,7,8,9)(11,13,16,12)(14,26,28,25)(15,20,32,23)(17,24,18,21)(19,30,22,29)(27,38,40,39)(31,33,36,34)(35,42,37,41);(11,14,16,28)(12,19,13,22)(15,27,32,40)(17,30,18,29)(20,35,23,37)(21,26,24,25)(31,39,36,38)(33,42,34,41);(4,8)(5,10)(11,15,16,32)(12,20,13,23)(14,27,28,40)(17,34,18,33)(19,35,22,37)(21,31,24,36)(25,38,26,39)(29,42,30,41)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408332);