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×#58542 |
| 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 | |
4598 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);(3,4,8,6)(5,9,7,10)(11,12,15,18)(13,19,20,16)(14,17,22,24)(21,25,26,23)(27,28)(29,36)(30,33)(31,39)(32,35)(34,40)(37,41)(38,42);(4,6)(5,7)(13,20)(16,19)(21,26)(23,25)(27,29,32,30)(28,33,35,36)(31,37,40,38)(34,41,39,42);(3,5,8,7)(4,9,6,10)(11,13)(12,16)(14,21)(15,20)(17,23)(18,19)(22,26)(24,25)(27,30,32,29)(28,33,35,36)(31,38,40,37)(34,41,39,42);(3,6,8,4)(5,9,7,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,26)(27,28)(29,36)(30,33)(31,34)(32,35)(37,42)(38,41)(39,40);(3,7,8,5)(4,9,6,10)(11,14,15,22)(12,17,18,24)(13,21,20,26)(16,23,19,25)(27,31,32,40)(28,34,35,39)(29,37,30,38)(33,41,36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011399);