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×#41791 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4486 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);(4,8)(9,10)(11,12,16,14)(13,19,20,17)(15,18,23,22)(21,25,26,24)(27,28,33,32)(29,36,30,34)(31,39,40,35)(37,41,38,42);(13,20)(15,23)(17,19)(18,22)(27,29,33,30)(28,34,32,36)(31,37,40,38)(35,41,39,42);(3,4)(5,9)(6,10)(7,8)(11,13)(12,17)(14,19)(15,21)(16,20)(18,24)(22,25)(23,26)(27,30,33,29)(28,34,32,36)(31,37,40,38)(35,42,39,41);(3,5,7,6)(4,9,8,10)(11,14,16,12)(13,17,20,19)(15,22,23,18)(21,24,26,25)(27,31,33,40)(28,35,32,39)(29,37,30,38)(34,41,36,42);(3,6,7,5)(4,10,8,9)(11,15,16,23)(12,18,14,22)(13,21,20,26)(17,24,19,25)(27,32,33,28)(29,34,30,36)(31,39,40,35)(37,41,38,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7546905);