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×#51874 |
| 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 | |
4562 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);(5,9)(8,10)(11,12)(13,19)(14,22)(15,16)(17,21)(18,23)(20,24)(25,26)(27,28,32,35)(29,30,33,37)(31,34,39,41)(36,38,40,42);(3,4,7,6)(5,8,9,10)(12,16)(17,19)(18,22)(24,25)(29,33)(30,37)(31,39)(34,41);(3,5,7,9)(4,8,6,10)(11,13,15,21)(12,17,16,19)(14,20,23,26)(18,24,22,25)(27,29)(28,30)(31,36)(32,33)(34,38)(35,37)(39,40)(41,42);(3,6,7,4)(5,10,9,8)(11,12)(13,17)(14,18)(15,16)(19,21)(20,24)(22,23)(25,26)(27,30)(28,33)(29,35)(31,38)(32,37)(34,40)(36,41)(39,42);(11,14)(12,18)(13,20)(15,23)(16,22)(17,24)(19,25)(21,26)(27,31,32,39)(28,34,35,41)(29,36,33,40)(30,38,37,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615021);