Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#36700 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| socle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4496 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)(5,8)(6,10)(7,9)(11,12)(13,17)(14,18)(15,16)(19,22)(20,24)(21,23)(25,26)(27,28,31,34)(29,32,36,40)(30,33,38,42)(35,39,37,41);(3,5,7,6)(4,8,9,10)(12,16)(17,22)(18,23)(24,26)(27,29,31,36)(28,32,34,40)(30,37,38,35)(33,41,42,39);(4,9)(8,10)(11,13)(12,17)(14,20)(15,19)(16,22)(18,24)(21,25)(23,26)(27,30)(28,33)(29,35)(31,38)(32,39)(34,42)(36,37)(40,41);(3,6,7,5)(4,10,9,8)(13,19)(17,22)(20,25)(24,26)(28,34)(30,38)(32,40)(35,37);(11,14,15,21)(12,18,16,23)(13,20,19,25)(17,24,22,26)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540607);