Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#57381 |
| 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 | |
4370 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);(11,12,15,18)(13,19,20,16)(14,17,22,24)(21,25,26,23)(27,28,32,33)(29,31,37,34)(30,38,39,35)(36,42,40,41);(3,4,7,9)(5,6,10,8)(13,20)(16,19)(21,26)(23,25)(28,33)(30,39)(31,34)(36,40);(3,5,7,10)(4,8,9,6)(11,13,15,20)(12,16,18,19)(14,21,22,26)(17,23,24,25)(27,29)(28,34)(30,40)(31,33)(32,37)(35,41)(36,39)(38,42);(3,5)(4,6)(7,10)(8,9)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,26)(27,30,32,39)(28,35,33,38)(29,36,37,40)(31,41,34,42);(3,6)(4,5)(7,8)(9,10)(11,14,15,22)(12,17,18,24)(13,21,20,26)(16,23,19,25)(27,31,32,34)(28,29,33,37)(30,41,39,42)(35,36,38,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008189);