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×#42703 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4,7,5)(6,9,10,8)(11,12,16,15)(13,17,21,20)(14,22,24,18)(19,26,23,25)(27,28,32,35)(29,33,36,31)(30,34,40,42)(37,41,38,39);(3,5,7,4)(6,9,10,8)(12,15)(17,20)(18,22)(25,26)(29,36)(31,33)(37,38)(39,41);(3,6,7,10)(4,8,5,9)(11,13,16,21)(12,17,15,20)(14,23,24,19)(18,26,22,25)(27,29)(28,33)(30,38)(31,35)(32,36)(34,39)(37,40)(41,42);(11,14,16,24)(12,18,15,22)(13,19,21,23)(17,25,20,26)(27,30)(28,34)(29,37)(31,39)(32,40)(33,41)(35,42)(36,38);(11,15,16,12)(13,20,21,17)(14,22,24,18)(19,26,23,25)(27,31,32,33)(28,29,35,36)(30,39,40,41)(34,37,42,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548489);