Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#30840 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1749 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);(3,4,7,10)(5,6,9,8)(11,12,15,21)(13,23,26,18)(14,28,30,36)(16,19,17,22)(20,31,38,32)(24,35,27,34)(25,40,29,42)(33,41,37,39);(3,5,7,9)(4,8,10,6)(11,13,15,26)(12,18,21,23)(14,29,30,25)(16,27,17,24)(19,35,22,34)(20,37,38,33)(28,40,36,42)(31,41,32,39);(3,6,7,8)(4,9,10,5)(12,19)(13,24)(16,17)(20,36)(21,22)(25,39)(26,27)(28,38)(29,41)(31,32)(34,35)(40,42);(4,10)(5,9)(11,14,15,30)(12,20,21,38)(13,25,26,29)(16,32,17,31)(18,33,23,37)(19,36,22,28)(24,39,27,41)(34,42,35,40)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409245);