Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#23367 |
| 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 | |
1630 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)(27,28,30,33,31,34,37,40)(29,32,36,39,35,38,41,42);(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)(28,32)(30,36)(31,35)(33,39)(34,38)(37,41)(40,42);(4,8)(7,9)(11,15,16,14)(12,18,19,24)(13,22,23,21)(17,25,20,26)(29,35)(32,38)(36,41)(39,42)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314840);