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×#50932 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4448 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);(3,4,7,5)(6,9,10,8)(11,12,16,17)(13,14,18,22)(15,23,24,19)(20,26,25,21)(27,28,32,35)(29,33,36,41)(30,31,34,40)(37,42,39,38);(3,5,7,4)(6,9,10,8)(12,17)(13,18)(15,24)(21,26)(29,36)(30,34)(31,40)(33,41);(3,6,7,10)(4,8,5,9)(11,13)(12,14)(15,20)(16,18)(17,22)(19,21)(23,26)(24,25)(27,29)(28,33)(30,39)(31,38)(32,36)(34,37)(35,41)(40,42);(11,14)(12,18)(13,17)(15,21)(16,22)(19,25)(20,23)(24,26)(27,30,32,34)(28,31,35,40)(29,37,36,39)(33,42,41,38);(11,15,16,24)(12,19,17,23)(13,20,18,25)(14,21,22,26)(27,31)(28,34)(29,38)(30,35)(32,40)(33,37)(36,42)(39,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613896);