Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46927 |
| 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 | |
4546 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);(4,7)(8,10)(11,12,16,14)(13,19,20,17)(15,18,23,22)(21,25,26,24)(27,28)(29,31)(30,34)(32,40)(33,36)(35,41)(37,42)(38,39);(13,20)(15,23)(17,19)(18,22)(27,29,33,30)(28,34,36,31)(32,37,41,38)(35,42,40,39);(3,4,6,7)(5,8,9,10)(11,13,16,20)(12,17,14,19)(15,21,23,26)(18,24,22,25)(27,30,33,29)(28,34,36,31)(32,38,41,37)(35,42,40,39);(3,5,6,9)(4,8,7,10)(11,14,16,12)(13,17,20,19)(15,22,23,18)(21,24,26,25)(27,31,33,34)(28,29,36,30)(32,39,41,42)(35,37,40,38);(11,15)(12,18)(13,21)(14,22)(16,23)(17,24)(19,25)(20,26)(27,32,33,41)(28,35,36,40)(29,37,30,38)(31,39,34,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7607970);