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×#55090 |
| 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 | |
4566 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);(11,12,15,16)(13,19,21,17)(14,22,23,18)(20,24,26,25)(27,28)(29,33)(30,34)(31,32)(35,37)(36,41)(38,40)(39,42);(3,4,7,8)(5,6,9,10)(12,16)(14,23)(17,19)(20,26)(28,31)(29,35)(30,38)(41,42);(4,8)(5,9)(11,13,15,21)(12,17,16,19)(14,20,23,26)(18,24,22,25)(27,29,32,35)(28,33,31,37)(30,39,38,36)(34,42,40,41);(3,5)(4,6)(7,9)(8,10)(11,12)(13,19)(14,18)(15,16)(17,21)(20,25)(22,23)(24,26)(27,30,32,38)(28,34,31,40)(29,36,35,39)(33,41,37,42);(3,6)(4,5)(7,10)(8,9)(11,14,15,23)(12,18,16,22)(13,20,21,26)(17,24,19,25)(27,31,32,28)(29,37,35,33)(30,40,38,34)(36,42,39,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009687);