Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#23368 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle, prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12)(13,20)(14,18)(15,24)(16,19)(17,23)(21,26)(22,25)(27,28,31,34,30,33,38,41)(29,35,37,42,36,32,39,40);(5,10)(7,9)(11,13,16,23)(12,17,19,20)(14,22,15,21)(18,26,24,25);(3,5,6,10)(4,7,8,9)(11,14,16,15)(12,18,19,24)(13,21,23,22)(17,25,20,26)(27,29,30,36)(28,32,33,35)(31,37,38,39)(34,40,41,42);(4,8)(7,9)(11,15,16,14)(12,18,19,24)(13,22,23,21)(17,25,20,26)(29,36)(32,35)(37,39)(40,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314841);