Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#30231 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1554 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)(5,7)(6,8)(9,10)(11,12,15,16)(13,17,20,23)(14,18,22,24)(19,25,21,26)(27,28,32,29,31,35,33,36)(30,38,41,39,40,37,42,34);(3,5,6,9)(4,7,8,10)(12,16)(17,23)(18,24)(25,26)(27,29,32,35,31,36,33,28)(30,39,41,37,40,34,42,38);(4,8)(7,10)(11,13,15,20)(12,17,16,23)(14,21,22,19)(18,26,24,25)(27,30,31,40)(28,34,35,39)(29,37,36,38)(32,42,33,41);(11,14,15,22)(12,18,16,24)(13,19,20,21)(17,25,23,26)(30,40)(34,39)(37,38)(41,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343530);