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×#57660 |
| 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 | |
4620 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)(5,9)(11,12)(13,18)(14,19)(15,23)(16,17)(20,24)(21,22)(25,26)(27,28,32,36)(29,34,30,33)(31,39,40,35)(37,41,38,42);(4,7)(8,10)(12,17)(14,18)(20,23)(22,25)(27,29,32,30)(28,33,36,34)(31,38,40,37)(35,42,39,41);(3,4,6,7)(5,8,9,10)(11,13,16,19)(12,18,17,14)(15,21,24,26)(20,25,23,22)(27,30,32,29)(28,34,36,33)(31,38,40,37)(35,42,39,41);(3,5,6,9)(4,8,7,10)(11,14,16,18)(12,19,17,13)(15,22,24,25)(20,26,23,21)(27,28)(29,34)(30,33)(31,35)(32,36)(37,42)(38,41)(39,40);(11,15,16,24)(12,20,17,23)(13,21,19,26)(14,22,18,25)(27,31,32,40)(28,35,36,39)(29,37,30,38)(33,41,34,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009092);