Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#32854 |
| 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×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1342 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);(4,5)(8,9)(11,12,15,18)(13,19,21,16)(14,17,23,26)(20,25,22,24)(27,28)(29,32)(30,34)(31,33)(35,39)(36,41)(37,40)(38,42);(3,4,7,5)(6,8,10,9)(11,13,15,21)(12,16,18,19)(14,22,23,20)(17,25,26,24)(27,29)(28,32)(30,35)(31,37)(33,40)(34,39)(36,38)(41,42);(3,5,7,4)(6,8,10,9)(14,23)(17,26)(20,22)(24,25)(29,35)(30,38)(32,39)(34,42)(36,37)(40,41);(3,6,7,10)(4,8,5,9)(11,14,15,23)(12,17,18,26)(13,20,21,22)(16,24,19,25)(28,33)(32,40)(34,42)(39,41)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538782);