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×#57424 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(11,12,15,18)(13,19,21,16)(14,17,22,24)(20,25,26,23)(27,28)(29,34)(30,35)(31,39)(32,33)(36,40)(37,38)(41,42);(3,4,7,9)(5,6,10,8)(14,22)(17,24)(20,26)(23,25)(28,33)(30,34)(36,39)(38,41);(3,5,7,10)(4,8,9,6)(11,13,15,21)(12,16,18,19)(14,20,22,26)(17,23,24,25)(27,29,32,35)(28,34,33,30)(31,37,40,42)(36,41,39,38);(3,5)(4,6)(7,10)(8,9)(11,12)(13,19)(14,17)(15,18)(16,21)(20,25)(22,24)(23,26)(27,30,32,34)(28,35,33,29)(31,38,40,41)(36,42,39,37);(3,6)(4,5)(7,8)(9,10)(11,14,15,22)(12,17,18,24)(13,20,21,26)(16,23,19,25)(27,31,32,40)(28,36,33,39)(29,37,35,42)(30,38,34,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007870);