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×#41853 |
| 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 | |
4600 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);(3,4,7,10)(5,8,6,9)(11,12)(13,14)(15,19)(16,17)(18,22)(20,21)(23,24)(25,26)(27,28,32,31)(29,33,37,36)(30,38,40,34)(35,42,39,41);(3,5,7,6)(4,8,10,9)(12,17)(13,18)(19,24)(20,25)(28,31)(33,36)(34,38)(41,42);(3,4)(5,8)(6,9)(7,10)(11,13,16,18)(12,14,17,22)(15,20,23,25)(19,21,24,26)(27,29,32,37)(28,33,31,36)(30,39,40,35)(34,42,38,41);(3,5,7,6)(4,8,10,9)(11,14)(12,18)(13,17)(15,21)(16,22)(19,25)(20,24)(23,26)(27,30)(28,34)(29,35)(31,38)(32,40)(33,41)(36,42)(37,39);(3,6,7,5)(4,9,10,8)(11,15,16,23)(12,19,17,24)(13,20,18,25)(14,21,22,26)(27,31,32,28)(29,36,37,33)(30,38,40,34)(35,42,39,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547164);