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×#50381 |
| 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);(3,4,6,8)(5,7,9,10)(12,16)(17,20)(18,23)(24,26)(27,28)(29,36)(30,35)(31,34)(32,33)(37,42)(38,41)(39,40);(12,16)(13,19)(14,22)(24,26)(27,29,33,31)(28,34,32,36)(30,38,40,37)(35,42,39,41);(3,4)(5,7)(6,8)(9,10)(11,12,15,16)(13,20,19,17)(14,18,22,23)(21,26,25,24)(27,30,33,40)(28,35,32,39)(29,37,31,38)(34,41,36,42);(11,13)(12,17)(14,21)(15,19)(16,20)(18,24)(22,25)(23,26)(27,31,33,29)(28,34,32,36)(30,37,40,38)(35,42,39,41);(3,5,6,9)(4,7,8,10)(11,14,15,22)(12,18,16,23)(13,21,19,25)(17,24,20,26)(27,32,33,28)(29,34,31,36)(30,39,40,35)(37,41,38,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613366);