Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#50508 |
| 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 | |
4398 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);(4,8)(9,10)(11,12,16,19)(13,15,18,24)(14,17,21,25)(20,23,22,26)(27,28)(29,35)(30,38)(31,34)(32,37)(33,39)(36,40)(41,42);(13,18)(14,21)(15,24)(17,25)(27,28,31,34)(29,32,37,35)(30,33,39,38)(36,40,42,41);(3,4)(5,9)(6,10)(7,8)(11,13,16,18)(12,15,19,24)(14,22,21,20)(17,26,25,23)(27,29)(28,32)(30,36)(31,37)(33,40)(34,35)(38,41)(39,42);(3,5,7,6)(4,9,8,10)(11,14)(12,17)(13,20)(15,23)(16,21)(18,22)(19,25)(24,26)(27,28)(29,32)(30,33)(31,34)(35,37)(36,40)(38,39)(41,42);(3,6,7,5)(4,10,8,9)(11,15)(12,18)(13,19)(14,23)(16,24)(17,22)(20,25)(21,26)(27,30,31,39)(28,33,34,38)(29,36,37,42)(32,40,35,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613045);