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×#51273 |
| 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 | |
4648 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);(5,9)(8,10)(11,12,16,20)(13,14,18,17)(15,19,23,26)(21,22,25,24)(27,28)(29,33)(30,34)(31,32)(35,37)(36,41)(38,42)(39,40);(3,4,7,6)(5,8,9,10)(13,18)(14,17)(15,23)(19,26)(28,31)(29,35)(34,39)(36,38);(3,5)(4,8)(6,10)(7,9)(11,13)(12,17)(14,20)(15,21)(16,18)(19,24)(22,26)(23,25)(27,29,32,35)(28,33,31,37)(30,38,40,36)(34,42,39,41);(11,14,16,17)(12,18,20,13)(15,22,23,24)(19,25,26,21)(27,30,32,40)(28,34,31,39)(29,36,35,38)(33,41,37,42);(3,6,7,4)(5,10,9,8)(11,15,16,23)(12,19,20,26)(13,21,18,25)(14,22,17,24)(27,31,32,28)(29,37,35,33)(30,39,40,34)(36,42,38,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614445);