Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#49449 |
| 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 | |
4686 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);(4,8)(9,10)(11,12,16,14)(13,19,21,17)(15,22,23,18)(20,24,26,25)(27,28)(29,30)(31,39)(32,33)(34,40)(35,38)(36,42)(37,41);(12,14)(17,19)(18,22)(24,25)(28,33)(29,35)(31,40)(37,42);(3,4,7,8)(5,9,6,10)(11,13)(12,17)(14,19)(15,20)(16,21)(18,24)(22,25)(23,26)(27,29,32,35)(28,30,33,38)(31,36,40,41)(34,37,39,42);(3,5,7,6)(4,9,8,10)(11,14,16,12)(13,19,21,17)(15,22,23,18)(20,25,26,24)(27,30)(28,29)(31,37)(32,38)(33,35)(34,36)(39,41)(40,42);(3,6,7,5)(4,10,8,9)(11,15,16,23)(12,18,14,22)(13,20,21,26)(17,24,19,25)(27,31)(28,34)(29,36)(30,37)(32,40)(33,39)(35,41)(38,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7611427);