Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#41601 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4,7,6)(5,9,10,8)(11,12,16,15)(13,17,21,20)(14,22,24,18)(19,26,23,25)(27,28)(29,34)(30,38)(31,36)(32,33)(35,40)(37,39)(41,42);(3,4,7,6)(5,8,10,9)(12,15)(17,20)(18,22)(25,26)(28,33)(29,36)(35,38)(37,42);(3,5,7,10)(4,8,6,9)(11,13,16,21)(12,17,15,20)(14,23,24,19)(18,26,22,25)(27,29,32,36)(28,34,33,31)(30,37,40,42)(35,41,38,39);(3,6,7,4)(5,9,10,8)(11,14,16,24)(12,18,15,22)(13,19,21,23)(17,25,20,26)(27,30,32,40)(28,35,33,38)(29,37,36,42)(31,39,34,41);(11,15,16,12)(13,20,21,17)(14,22,24,18)(19,26,23,25)(27,31,32,34)(28,36,33,29)(30,39,40,41)(35,42,38,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547324);