Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#37574 |
| 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 | |
4578 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)(12,16)(14,22)(17,19)(20,26)(27,28,31,32)(29,35,37,33)(30,34,38,39)(36,41,42,40);(3,4,7,6)(5,8,9,10)(12,16)(17,19)(18,23)(24,25)(28,32)(33,35)(34,39)(40,41);(3,5)(4,8)(6,10)(7,9)(11,12)(13,19)(14,18)(15,16)(17,21)(20,25)(22,23)(24,26)(27,28)(29,33)(30,34)(31,32)(35,37)(36,40)(38,39)(41,42);(3,6,7,4)(5,10,9,8)(11,13,15,21)(12,17,16,19)(14,20,22,26)(18,24,23,25)(27,29,31,37)(28,33,32,35)(30,36,38,42)(34,40,39,41);(11,14)(12,18)(13,20)(15,22)(16,23)(17,24)(19,25)(21,26)(27,30,31,38)(28,34,32,39)(29,36,37,42)(33,40,35,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541019);